Jules Chenal
email: jules.chenal@univ-lille.fr

Poincaré Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces

Keywords: [Nik80], 16 c, f, [Kha75], 17 c, f, [Kha75] Remarks on the Homomorphisms j * p, 136, 18 c, f, Definition 3, 3, 19 c, f, Definition 4, 35

In Bounding the Betti Numbers of Real Hypersurfaces near the Tropical Limit, [RS23], A. Renaudineau and K. Shaw presented a filtration of the cellular complex of a real, primitively patchworked, hypersurface in a non-singular toric variety. This filtration, inspired by the work of Kalinin, allows, by computation of the first page of its induced spectral sequence, for individual bounds on the Betti numbers of such a hypersurface. We present two structural results about this spectral sequence. The first is a Poincaré duality satisfied by all its pages of positive index. The second is a vanishing criterion. It reformulates the vanishing of the boundary operators of the spectral sequence in terms of homological properties of the inclusion of the hypersurface in its surrounding toric variety. It implies further that the Renaudineau-Shaw spectral sequence of a T-hypersurface degenerates at the second page if and only if the T-hypersurface satisfies a real version of the Lefschetz hyperplane section theorem. As a consequence the spectral sequence of an odd degree T-hypersurface in a projective space always degenerates at the second page.

Introduction

Let t * (R) be a n-dimensional real vector space endowed with a lattice t * (Z). A smooth polytope of t * (R) is a full dimensional polytope P whose vertices lie in the lattice t * (Z) and whose associated toric variety Y is non-singular. The real locus Y (R) of such toric variety can be given a regular CW-complex structure RP obtained by gluing 2 n copies of P along their faces. This CW-complex structure is invariant under the action of the compact real algebraic torus Hom Z (t * (Z); {±1}) ∼ = t(F 2 ). Any primitive triangulation K of P can be uniquely lifted as an invariant subdivision RK of RP . This subdivision might fail to be a triangulation but is always a ∆-complex 1 . A classical construction 2 associates a pure sub-complex of codimension 1 of the barycentric subdivision of RK to any closed cochain α ∈ Z 1 (RK; F 2 ). This "hypersurface" determines an (n -1)-cycle whose homology class is Poincaré dual to the cohomology class of α. When the cochain α observes some symmetries with respect to the action of t(F 2 ) it is necessarily cohomologous to a canonical cocycle ω RX and the associated hypersurface is called a T-hypersurface. Such hypersurfaces are parametrised by cochains ε ∈ C 0 (K; F 2 ) called sign distributions. They are always PL-smooth 3 and we denote them by RX ε . When K is convex 4 , O. Viro's Patchworking Theorem 5 asserts that RX ε is isotopic inside Y (R) to the real locus of a non-singular algebraic hypersurface of Y . This theorem motivated the study of such hypersurfaces. A. Renaudineau and K. Shaw used a version of I.O. Kalinin's spectral sequence 6 (E r p,q (RX ε )) p,q,r∈N to derive upper bounds 7 on the Betti numbers of RX ε in terms of quantities associated with the triangulation. When K is convex, these quantities are the tropical Hodge numbers 8 of the tropical hypersurface dual to K. They correspond 9 , regardless of the convexity of K, to the Hodge numbers of the zero locus of a non-singular section of the line bundle of Y associated with the polytope P . These inequalities, when added together, specilise to the Smith-Thom inequality 10 .

As pointed out by E. Brugallé, L. Lopez de Medrano, and J. Rau in [START_REF] Brugallé | Combinatorial Patchworking: Back from Tropical Geometry[END_REF], almost none of these results about RX ε depends on the convexity of the triangulation K. Here we study the spectral sequence (E r p,q (RX ε )) p,q,r∈N and its dual (E p,q r (RX ε )) p,q,r∈N for an arbitrary primitive triangulation K, and tackle the following conjecture of A. Renaudineau and K. Shaw: Conjecture (A. Renaudineau and K. Shaw, Conjecture 1.10, [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF]). The spectral sequence (E r p,q (RX ε )) p,q,r∈N degenerates at the second page.

In a first time we prove a general lemma about the Poincaré duality in spectral sequences:

Lemma 1.5 (Chain Reaction). Let (A; d) be an increasingly filtered, graded, differential algebra of finite dimension over the field F and E p,q r (A) p,q,r≥0 denote its associated spectral sequence. If there are integers r 0 ≥ 1, m, n ≥ 0 for which:

1. The vector space E p,q r 0 (A) vanishes whenever p > m or q > n;

2. The vector space E m,n r 0 (A) has dimension 1;

3. The bilinear pairing E p,q r 0 (A) ⊗ F E m-p,n-q r 0 (A) → E m,n r 0 (A) is non-degenerate, then for all r ≥ r 0 :

1. The vector space E p,q r (A) vanishes whenever p > m or q > n;

2. The vector space E m,n r (A) has dimension 1;

3. The bilinear pairing E p,q r (A) ⊗ F E m-p,n-q r (A) → E m,n r (A) is non-degenerate.

Then we describe the dual filtration in cohomology and deduce from its properties that the dual spectral sequence is a spectral ring. Using this structure of algebra as well as the tropical Poincaré duality11 for the hypersurface X dual to K we show that this sequence is symmetric: Theorem 4.27 (Symmetry). For all r ≥ 1, the E r -pages of the Renaudineau-Shaw spectral sequence computing the cohomology of RX ε satisfy the Poincaré duality. That is to say:

1. The vector space E p,q r (RX ε ) vanishes whenever p > n -1 or q > n -1;

2. The vector space E n-1,n-1 r (RX ε ) has dimension 1;

3. The bilinear pairing ∪ :

E p,q r (RX ε ) ⊗ E n-1-p,n-1-q r (RX ε ) → E n-1,n-1 r (RX ε ) is non- degenerate.
In particular, E p,q r (RX ε ) is isomorphic to E n-1-p,n-1-q r (RX ε ) and d p,q r has the same rank and kernel dimension as d n-1-p+r,n-2-q r . Using this symmetry and the tropical Lefschetz Hyperplane Section Theorem12 we are able to topologically characterise the vanishing of some differentials of this spectral sequence: Theorem 4.30 (Vanishing Criterion). Let r ≥ 2 be an integer congruent to n modulo 2. The differentials of the relevant page E r (RX ε ) are all zero if and only if i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective when q equals n-r 2 .

And we derive the corollary:

Corollary 4.31 (Criterion of Degeneracy). Let r 0 ≥ 2 be an integer congruent to n modulo 2. The differentials d p,q r of the Renaudineau-Shaw spectral sequence computing the cohomology of RX ε vanish for all p, q ≥ 0, and all r ≥ r 0 , if and only if the cohomological inclusions i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) are injective for all q ≤ n-r 0 2 . In particular, all the differentials d p,q r are trivial for all r ≥ 2 if and only if the cohomological inclusions i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) are injective for all q ≤ ⌊ n 2 ⌋ -1.

It can be reformulated as the comparison of two quantities associated with the pair RX ε ⊂ RP . The first one is the following: Définition 4.32. We define the degeneracy index of RX ε as: r(RX ε ) := min{r 0 ≥ 0 | d p,q r = 0, ∀p, q ∈ N, ∀r ≥ r 0 }.

And the second one is:

Définition 4.33. The rank of RX ε is defined as:

ℓ(RX ε ) := max{q 0 ≥ 0 | i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective for all q ≤ q 0 }.

Using these definitions Corollary 4.31 can be expressed by two inequalities:

Corollary 4.34. We have the inequalities:

ℓ(RX ε ) ≥ n -r(RX ε ) 2 ,
with equality if r(RX ε ) ≥ 3 + 1-(-1) n 2 and:

r(RX ε ) ≤ max (2; n -2ℓ(RX ε ) -1) , with equality if ℓ(RX ε ) ≤ n-5 2 . The quantity ℓ was previously introduced for subsets of real projective spaces by I.O. Kalinin. "The rank is involved in many restrictions on topology of real algebraic hypersurfaces of a given degree" 13 . It was studied by I.O. Kalinin 14 , V. Nikulin 15 , or V. Kharlamov 16 for real projective hypersurfaces. Let V be a real algebraic hypersurface of the projective space P n then ℓ(V (R)) = n -1 if V is of odd degree and ℓ(V (R)) ≤ n 2 if V is of even degree 17 . The same holds for ℓ(RX ε ) even when K is non-convex. As a consequence, the spectral sequences (E r p,q (RX ε )) p,q,r∈N and (E p,q r (RX ε )) p,q,r∈N both degenerate at the second page when P is an odd dilate of a primitive simplex.

Corollary 4.40. The Renaudineau-Shaw spectral sequence of a T-hypersurface of odd degree in a projective space degenerates at the second page.

We generalise this property to a broader variety of polytopes. We define the degree 18 of RX ε as the cohomology class [ω RX ] ∈ H 1 (RP ; F 2 ), and a number 19 ι[ω RX ] that satisfies: Proposition 4.39. We have the inequality:

ℓ(RX ε ) ≥ ι[ω RX ].
In particular it follows from Corollary 4.38 that if ι[ω RX ] ≥ n 2 -1 then all the differentials of the pages (E p,q r (RX ε )) p,q∈N vanish for all r ≥ 2. However, this proposition has some limitations since, in a cube, ι[ω RX ] is always 0 for all primitive triangulations K. In the last section we give the construction of particular triangulations (V n d ) n,d≥1 of the d-dilates of primitive n-simplices for which we prove:

Theorem 5.9 (Rank Maximality). If K is a Viro triangulation (V n d ) n,d≥1 of the corresponding dilatation of the simplex P n then the homological inclusion:

i q : H q (RX ε ; F 2 ) → H q (RP n ; F 2 ),
is surjective for all q ≤ n-1 2 and ε ∈ C 0 (K; F 2 ). That is to say, a projective hypersurface obtained from a primitive patchwork on the Viro triangulation always has maximal rank.

Which implies the corollary:

Corollary 5.10. The Renaudineau-Shaw spectral sequences computing the homology and the cohomology of the hypersurface RX ε ⊂ RP n constructed from a Viro triangulation K ∈ (V n d ) n,d≥1 and a sign distribution ε ∈ C 0 (K; F 2 ) degenerates at the second page.

A. Renaudineau and K. Shaw Conjecture can be rephrased as r(RX ε ) ≤ 2 or equivalently ℓ(RX ε ) ≥ n 2 -1. However, as in the case of Theorem 5.9, we believe the stronger statement ℓ(RX ε ) ≥ n-1 2 might even be true in full generality. 1 Duality in Spectral Sequences

Let F be a field.

Definition 1.1. An increasingly filtered, graded, differential algebra of finite dimension over the field F is an associative F-algebra A of finite dimension over F together with:

1. A graduation A = q≤0 A q compatible with the product in the following way: for all α ∈ A q and all α ′ ∈ A q ′ , αα ′ equals (-1) qq ′ α ′ α and belongs to A q+q ′ ;

2. An increasing filtration F 0 A ⊂ F 1 A ⊂ ... ⊂ A compatible with the product: for all p, p ′ ∈ N, F p AF p ′ A is included in F p+p ′ A;

A differential d :

A → A which is a morphism of graded vector spaces of degree 1, whose square vanishes, that is compatible with the filtration dF p A ⊂ F p A, for all p ∈ N, and for which the Leibniz rule holds: for all α ∈ A q and α ′ ∈ A q ′ , d(αα ′ ) = (dα)α ′ + (-1) q αdα ′ .

Such an object is in particular a finite dimensional cochain complex. The compatibility of the filtration with the differential allows us to use the techniques of spectral sequences to compute the cohomology of A. We denote the spectral sequence associated with the filtration (F p A) p∈N by E p,q r (A) p,q,r∈N . We adopt the following index convention. For all integers p, q, r ≥ 0:

E p,q r (A) := Z p,q r (A) + F p-1 A q dZ p+r-1,q-1 r-1 (A) + F p-1 A q , with: Z p,q r (A) := {α ∈ F p A q | dα ∈ F p-r A q+1 } , and Z p,q
-1 (A) := 0. In this setting the differential d r of the r-th page has bidegree (-r; +1) and is the factorisation of the restriction of d. By definition of the objects we consider here, we deal with first quadrant spectral sequences arising from bounded filtrations, hence, the spectral sequence E p,q r (A) p,q,r≥0 converges20 toward the cohomology of A. The compatibility of the filtration with the product ensures that E p,q r (A) p,qr∈N is even a spectral ring. It means that there is a well defined product:

E p,q r (A) ⊗ F E p ′ ,q ′ r (A) → E p+p ′ ,q+q ′ r (A) ,
which satisfies, for all α ∈ E p,q r (A) and α ′ ∈ E p ′ ,q ′ r (A), the graded commutativity:

αα ′ = (-1) qq ′ α ′ α ,
and Leibniz rule:

d p+p ′ ,q+q ′ r (αα ′ ) = d p,q r (α)α ′ + (-1) q αd p ′ ,q ′ r (α ′ ) .
These constructions are explained in §2.3. Spectral sequences of algebras pp.44-46 of [START_REF] Mccleary | A User's Guide to Spectral Sequences[END_REF].

Although we should emphasised that we did not adopt the same index convention as J. Mc-Cleary. We justify our choice by the degree of generality of the objects we discuss here. Since we do not have much context about the arising of the spectral sequence we do not find meaningful to perform the change of index [p → p + q] often used with double complexes for which it is particularly adapted.

Definition 1.2. Let A be an increasingly filtered, graded, differential algebra of finite dimension over the field F. We denote by A * the dual chain complex. By the Universal Coefficient Theorem21 , the homology of A * is dual to the cohomology of A.

P. Deligne gave a definition of the dual filtration of a filtered object of an Abelian category. It is a filtration of the same object in the opposite category. It is defined in such a way that the graded objects of the dual filtration are dual to the graded objects of the initial filtration. This construction is designed to be used with contravariant functors. Definition 1.3 (P. Deligne, [START_REF] Deligne | Théorie de Hodge[END_REF] (1.1.6) p.7). Let C be an Abelian category, we denote by C op the opposite category and by (-) op : (C op ) op → C the "identical contravariant functor". Let V (resp. W ) be an object of C endowed with an increasing (resp. decreasing) filtration V (k) k∈Z (resp. W (k) k∈Z ). The dual decreasing (resp. increasing) filtration of V op (resp. W op ), is defined, for all k ∈ Z, by the formulae:

(V op ) (k) := im V ↠ V V (k-1) op , respectively: (W op ) (k) := im W ↠ W W (k+1) op , so that V (k) / V (k-1) op is naturally isomorphic 22 to (V op ) (k)/(V op ) (k+1) .
If C is the category of vector spaces for instance (or even of cochain complexes) over a field F, applying the usual duality functor Hom F (-; F) to this abstract construction yields a filtration, in the usual sense, of opposite growth of the dual vector space (or chain complex).

Following this definition the dual chain complex A * is naturally endowed with a dual decreasing filtration of chain complexes 0 ⊂ ... ⊂ F 1 A * ⊂ F 0 A * = A * . We denote by E r p,q (A * ) r,p,q≥0 the associated spectral sequence. It is defined for all integers p, q, r ≥ 0, by:

E r p,q (A * ) := Z r p,q (A * ) + F p+1 A * q ∂Z r-1 p-r+1,q+1 (A * ) + F p+1 A * q ,
with ∂ = Hom F (d; F), and:

Z r p,q (A * ) := {a ∈ F p A * q | ∂a ∈ F p+r A * q-1 } ,
and the convention Z -1 p,q (A * ) := 0. The induced boundary operators ∂ r have bidegree (+r; -1).

Proposition 1.4. The spectral sequences E r p,q (A * ) p,q,r≥0 and E p,q r (A) p,q,r≥0 are dual of each other through a collection of duality pairings:

⟨-; -⟩ : E p,q r (A) ⊗ F E r p,q (A * ) → F ,
for all integers p, q, r ≥ 0, and for which d p,q r is the adjoint of ∂ r p-r,q+1 .

Proof. Let q ∈ N, and denote by ⟨-; -⟩ : A q ⊗ A * q → F the usual duality pairing. By definition d and ∂ are adjoints of each other i.e. ⟨dα; a⟩ = ⟨α; ∂a⟩ for all α ∈ A q and all a ∈ A * q+1 . Since F p+1 A * q is the image in A * q of ( A q / F p A q ) * , it is precisely the vector space {a ∈ A * q | ⟨α ; a⟩ = 0, ∀α ∈ F p A q }. It follows that the restriction of ⟨-; -⟩ to the sum of the sub-spaces:

Z p,q r (A) + F p-1 A q ⊗ F ∂Z r-1 p-r+1,q+1 (A * ) + F p+1 A * q ,
and: dZ p+r-1,q r-1

(A) + F p-1 A q ⊗ F Z r p,q (A * ) + F p+1 A * q ,
vanishes. Thus ⟨-; -⟩ factors through the quotient map to give rise to a well define bilinear product:

⟨-; -⟩ : E p,q r (A) ⊗ F E r p,q (A * ) → F . The morphisms (d p,q r ) p,q,r∈N and (∂ r p,q ) p,q,r∈N are factorisations of d and ∂ respectively. Hence, they are adjoints of each others for the new pairings. By definition of the dual filtration, the pairings ⟨-; -⟩ defined on E p,q 0 (A) ⊗ F E 0 p,q (A * ) are non-degenerate for all p, q ∈ N. Moreover, the fundamental property of spectral sequences, the one that enables us to compute a page as the cohomology (or homology) of the preceding, allows us to repetitively use the Universal Coefficients Theorem23 and deduce that all the pairings are non-degenerate. We should emphasise that we only considered finite dimensional chain and cochain complexes of vector spaces.

Lemma 1.5 (Chain Reaction). Let (A; d) be an increasingly filtered, graded, differential algebra of finite dimension over the field F and E p,q r (A) p,q,r≥0 denote its associated spectral sequence. If there are integers r 0 ≥ 1, m, n ≥ 0 for which:

1. The vector space E p,q r 0 (A) vanishes whenever p > m or q > n;

2. The vector space E m,n r 0 (A) has dimension 1;

3. The bilinear pairing E p,q r 0 (A)

⊗ F E m-p,n-q r 0 (A) → E m,n r 0 (A) is non-degenerate,
then for all r ≥ r 0 :

1. The vector space E p,q r (A) vanishes whenever p > m or q > n;

2. The vector space E m,n r (A) has dimension 1;

3. The bilinear pairing E p,q r (A)

⊗ F E m-p,n-q r (A) → E m,n r (A) is non-degenerate.
Proof. By recursion we only need to prove the statement for r 0 + 1. The first assertion is a consequence of E p,q r 0 +1 (A) being a sub-quotient of E p,q r 0 (A). The second follows from the isomorphism between E m,n r 0 +1 (A) and the (m, n)-cohomology group of d r 0 . Indeed, around E n,m r 0 the cochain complex is the following:

E m+r 0 ,n-1 r 0 (A) E m,n r 0 (A) E m-r 0 ,n+1 r 0 (A) . d m+r 0 ,n-1 r 0 d m,n r 0
By assumption, both the left hand group and the right hand group vanish. Hence, E m,n r 0 +1 (A) is isomorphic to E m,n r 0 (A) and has dimension 1. For the last part we chose an isomorphism : E m,n r 0 (A) → F and consider the musical isomorphism:

D p,q r 0 : E p,q r 0 (A) → E m-p,n-q r 0 (A) * ∼ = E r 0 m-p,n-q (A * ) ,
defined by the formula α D p,q r 0 (β) = αβ for all α ∈ E m-p,n-q r 0 (A) and β ∈ E p,q r 0 (A). This is, up to sign, a (co)-chain complex isomorphism since: α ; D p-r 0 ,q+1 r 0 (d p,q r 0 β) = α d p,q r 0 β = (-1) q d m-p+r 0 ,n-q-1 r 0 α β = (-1) q α ; ∂ r 0 m-p,n-q D p,q r 0 (β) .

It induces, nonetheless, an isomorphism (D p,q r 0 ) * between E p,q r 0 +1 (A) and E r 0 +1 m-p,n-q (A * ). We observe now that the isomorphism between E m,n r 0 +1 (A) and E m,n r 0 (A) induces an isomorphism : E m,n r 0 +1 (A) → F such that for any two α ∈ E m-p,n-q r 0 +1

(A) and β ∈ E p,q r 0 +1 (A), respectively represented by the two cocyles α ′ ∈ E m-p,n-q r 0 (A) and β ′ ∈ E p,q r 0 (A), we have:

αβ = α ′ β ′ .
Therefore, the bilinear pairing constructed from the product and the integration:

E m-p,n-q r 0 +1 (A) ⊗ F E p,q r 0 +1 (A) → F,
has its musical morphism:

D p,q r 0 +1 : E p,q r 0 +1 (A) -→ E r 0 +1 m-p,n-q (A * ) ,
induced in cohomology by D p,q r 0 . It is an isomorphism of chain complexes, so the third assertion follows.

A direct consequence of this lemma is the symmetry of the pages of the spectral sequence following the E r 0 -page.

Proposition 1.6. Let (A; d) be a increasingly filtered, graded, differential algebra of finite dimension over the field F and E p,q r (A); d p,q r p,q,r≥0 denote its associated spectral sequence. If the spectral sequence of A satisfies the Poincaré duality at the r 0 -th page then for all r ≥ r 0 , and all p, q ≥ 0:

D p,q r : E p,q r (A) ∼ =
-→ E r m-p,n-q (A * ) , and:

d p,q r = (-1) q (D p-r,q+1 r ) -1 • (d m-p+r,n-q-1 r ) * • D p,q
r . In particular, E p,q r (A) and E m-p,n-q r (A) have the same dimension and d p,q r and d m-p+r,n-q-1 r have same rank and kernel dimension.

Lemma 1.7. Let W 0 ⊂ V 0 be two finite dimensional vector spaces over F. Let V 1 be a sub-space of V 0 , and

W 1 denote V 1 ∩ W 0 . Then the image of V ⊥ 1 in W * 0 is W ⊥ 1 .
Proof. The inclusion of the image of V ⊥ 1 in W ⊥ 1 is direct. Let us show that we can obtain every form of W ⊥ 1 as a the restriction to W 0 of a form of V ⊥ 1 . We can write V 0 as the direct sum

W 1 ⊕ U ⊕ U ′ in such a way that W 1 ⊕ U is V 1 and W 1 ⊕ U ′ is W . If α ∈ W ⊥
1 we extend it by 0 on U to obtain a form in V ⊥ 1 that restricts to the original one.

Basic Objects and Notations

Notations. Let t * (Z) denote a free Abelian group of finite rank n ≥ 1, t(Z) denote its dual Hom Z (t * (Z); Z).For all commutative ring R, t * (R) and t(R) respectively denote t * (Z) ⊗ Z R and t(Z) ⊗ Z R.

Definition 2.1. Let P be a full dimensional polytope of t * (R). We say that P is integral if its vertices lie in the lattice t * (Z). For all Q faces of P , denoted by Q ≤ P , we denote its tangent space by T Q. If P is integral, the vector space T Q is rational for all Q ≤ P . We say that P is smooth if it is integral and if every set {v Q : dim Q = n -1 and p ∈ Q}, for p a vertex of P and v Q a generator of the lattice {v ∈ t(Z) | α(v) = 0, ∀α ∈ T Q}, is a basis of t(Z). The polytope P is smooth if and only if it is the moment polytope of a non-singular toric variety 24 .

Figure 1: The cubical subdivision of the triangle.

Definition 2.2. Let P be a smooth polytope of t * (R). A primitive triangulation K of P is a triangulation whose vertices are integral, i.e. lie in the lattice t * (Z) and whose n-simplices have minimal normalised lattice volume, i.e. 1 n! . For all p-simplex σ p of K we denote its tangent space by T σ p ⊂ t * (R), i.e. the vector direction of the affine space spanned by its vertices.

Definition 2.3. We say that a regular CW-complex K is a ∆-complex if all of its closed cells are isomorphic to simplices as CW-complexes.

Definition 2.4. The cubical subdivision of a ∆-complex K is a regular subdivision that sits between K and its barycentric subdivision. All of its open cells are made by recombining open barycentric simplices. Every pair σ p ≤ σ q of simplices of K corresponds to an open cell : the union of the open barycentric simplices whose indexing flags start with σ p and end with σ q . The name comes from the fact that every closed cell, as a CW-complex, has the combinatoric of a triangulation of a cube. An example of cubical subdivision is depicted in Figure 1 where the triangle is subdivided into three squares. The cube indexed by σ p ≤ σ q is a face of the cube indexed by σ p ′ ≤ σ q ′ if and only if σ p ′ ≤ σ p ≤ σ q ≤ σ q ′ . Definition 2.5. Let P be a smooth polytope endowed with a primitive triangulation K. The dual hypersurface X of K is the sub-complex of its cubical subdivision made of the cubes indexed by the pairs of simplices σ p ≤ σ q for which p ≤ 1. See Figure 2 for examples. Definition 2.6. Any CW-complex E can be seen as a partially ordered set and thus as a small category. If we denote this category by Cell E and R is a commutative ring, a cellular cosheaf of R-modules F on E is a contravariant functor:

F : Cell E op -→ Mod R .
The morphism associated with an adjacent pair e p ≤ e q , denoted by -| e q e p : F (e q ) → F (e p ), is called an extension morphism of F . Dually a cellular sheaf of R-modules G on K is a covariant functor:

G : Cell E -→ Mod R .
The morphism associated with an adjacent pair e p ≤ e q , denoted by -| e q e p : G(e p ) → G(e q ), is called a restriction morphism of G. If C is any category and one has two functors A, from Cell E to C, and B, from C to Mod R , one gets either a cellular sheaf or a cellular cosheaf by composing A with B. If E is a ∆-complex, we call a cellular sheaf (resp. cosheaf) on its cubical subdivision a cubical sheaf (resp. cubical cosheaf ) on E. Definition 2.7 (Morphisms of Cellular Sheaves and Cosheaves). A morphism of cellular sheaves (resp. cosheaves) f : F → F ′ is a natural transformation. Such morphism is said to be injective (resp. surjective, resp. invertible) if the associated morphisms f e : F (e) → F ′ (e) are injective (resp. surjective, resp. invertible) for all cells e. The kernel, image, and cokernel of such morphism f are the "cell-wise" kernel, image, and cokernel. They are sheaves (resp. cosheaves) themselves with the induced restriction (resp. extension) morphisms because f is a natural transformation.

We give the definitions of the tropical sheaves and cosheaves associated with a primitive triangulation K of a smooth polytope P . All these objects usually take values in the category of Abelian groups 25 , however we only consider their reduction modulo 2 here, therefore we only set the notations for their reduced version.

Definition 2.8. Let P be a smooth polytope of t * (R) endowed with a primitive triangulation K. The first cellular cosheaf below is defined on K, all the other cellular sheaves and cosheaves are defined on the cubical subdivision of K.

1. The sedentarity cosheaf Sed : for a simplex σ p of K the group Sed(σ p ) is given by:

Sed(σ p ) := v ∈ t(F 2 ) | α(v) = 0, ∀α ∈ (T Q ∩ t * (Z)) ,
where Q is the smallest face of P containing the relative interior26 of σ p . If σ p ≤ σ q the extension morphism Sed(σ q ) → Sed(σ p ) is the inclusion; 2. The cubical subdivision of the sedentarity, that we denote by the same symbol Sed : for a cube of K indexed by a pair of simplices σ p ≤ σ q the group Sed(σ p ; σ q ) is given by: Sed(σ p ; σ q ) := Sed(σ q ), A cube indexed by σ p ≤ σ q is a face of the cube indexed by σ p ′ ≤ σ q ′ if and only if σ p ′ ≤ σ p ≤ σ q ≤ σ q ′ . In this case the extension morphism Sed(σ p ′ ; σ q ′ ) → Sed(σ p ; σ q ) is also given by the inclusion;

3. The cosheaf F P 1 is defined as the quotient:

F P 1 := t(F 2 ) / Sed ,
where t(F 2 ) is understood as the constant cosheaf;

4. The cosheaves F P p , for all p ∈ N, are the exterior powers of F P 1 :

F P p := p F P 1 ,
in particular F P 0 is the constant cosheaf F 2 ;

5. For all p ∈ N, the sheaf F p P is the dual of F P p , that is to say:

F p P := Hom F 2 (F P p ; F 2 );
6. For all k ∈ N, the groups of the cosheaf F X k are defined on cubes indexed by σ 1 ≤ σ q by the formula:

F X k (σ 1 ; σ q ) := k v ∈ t(F 2 ) Sed(σ 1 ; σ q ) | α(v) = 0 ,
where α is a generator of T σ 1 ∩ t * (Z). For a more general cube σ p ≤ σ q the group is given by:

F X k (σ p ; σ q ) := σ 1 ≤σ p F X k (σ 1 ; σ q ).
See Figure 3 for an example. The morphisms between these groups are given by quotients and inclusions. Note that even though F X k is defined on the whole cubical subdivision of K, its support 27 is contained in X. It is a sub-cosheaf of F P p and we denote the inclusion by i p : F X p ⊂ F P p ;

7. For all p ∈ N, the sheaf F p X is the dual of F X p , that is to say:

F p X := Hom F 2 (F X p ; F 2 ),
we denote the adjoint projection of i p by i p :

F p P → F p X . 0 0 Z∂ x Z∂ x = 0 Z∂ x 0 0 = Z(∂ x -∂ y ) Z(∂ x -∂ y ) Z(∂ x -∂ y ) Z 2
Figure 3: A triangle and the groups associated by F P 1 to some of its cubical cells.

Definition 2.9. Let σ p be a simplex of K. Its tangent space is a p-dimensional rational sub-space of t * (R). We denote by ω(σ p ) the generator of the line p (T

σ p ∩ t * (Z)) ⊗ Z F 2 .
We recall a basic construction of multilinear algebra in order to give a different definition of the cosheaves (F X p ) p≥0 . Definition 2.10 (Contraction). Let V be a finite dimensional vector space over a field F, and k, l ∈ N be integers. For all α ∈ l V * and v ∈ l+k V the contraction α • v is the only element of k V satisfying:

β(α • v) = (β ∧ α)(v),
for all β ∈ k V * . This construction is dual to the interior product.

Proposition 2.11. For all pairs of adjacent simplices σ p ≤ σ q of K, we have the exact sequence:

0 F X k (σ p ; σ q ) F P k (σ p ; σ q ) F P k-p (σ p ; σ q ). ω(σ p )•-
Proof. Let us denote by V the quotient of t(F 2 ) by Sed(σ p ; σ q ). For all simplices σ 1 ≤ σ p the linear form ω(σ 1 ) vanish on Sed(σ p ; σ q ). We abuse the notations and also denote by ω(σ 1 ) the factorisation of ω(σ 1 ) by the quotient map t(F 2 ) → V . From the sixth point of Definition 2.8 we have:

F X k (σ p ; σ q ) = σ 1 ≤σ q k ker(ω(σ 1 )) ⊂ k V.
27 i.e. the union of the closed cubes carrying non-trivial groups.

Let us choose a vertex σ 0 of σ p . Since σ p is primitive the set {ω(σ 1 ) : σ 0 ≤ σ 1 ≤ σ p } is, up to ordering, a basis of the vector space T σ p ∩ t * (Z) ⊗ Z F 2 ⊂ V * . We order this set {ω 1 , ..., ω p } and add to it some vectors α 1 , ..., α s ∈ V * in order to have a basis of V * . We denote by {e 1 , ..., e p , f 1 , ..., f s } the dual basis. We see that F X k (σ p ; σ q ) is spanned by the vectors28 e I ∧ f J for all I ⊂ {1, ..., p} and J ⊂ {1, ..., s} satisfying |I| + |J| = k and |I| < p. We can observe that ω(σ p ) = ω 1 ∧ ... ∧ ω p and that:

ω(σ p ) • e I ∧ f J = f J if I = {1, ..., p} 0 otherwise ,
from which the proposition follows.

Definition 2.12. Let F be a cellular cosheaf of F 2 -vector spaces on E a regular CW-complex, we denote by (C k (E; F ); ∂) k≥0 the chain complex of cellular chains with coefficients in F . For all k ≥ 0, its k-th group of chains is defined by the formula:

C k (E; F ) := e k ∈E F (e k ).
We write an element of C k (E; F ) the following way:

c = e k ∈E c e k ⊗ e k .
The boundary operator ∂ : : F (e k+1 ) → F (e k ) denotes the extension morphism of F . In a regular CWcomplex, if e k ≤ e k+2 then there are exactly two distinct e k+1 between e k and e k+2 and this ensures that ∂ • ∂ = 0. A morphism of cosheaves f : F → F ′ induces a morphism between the associated chain complexes. This associated morphism of chain complexes is injective, surjective, or invertible if and only f :

C k+1 (E; F ) → C k (E; F ) is
F → F ′ is. If dually G is a cellular sheaf of F 2 -
vector spaces on E, we denote by (C k (E; G); d) k≥0 the cochain complex of cellular cochains with coefficients in G. For all k ≥ 0, its k-th group of cochains is defined by the formula:

C k (E; G) := e k ∈E G(e k ).
The coboundary operator, or differential, d : : G(e k ) → G(e k+1 ) denotes the restriction morphism of G. As in the case of chain complexes we have d • d = 0. A morphism of sheaves f : F → F ′ induces a morphism between the associated chain complexes. This associated chain complexes morphism is injective, surjective, or invertible if and only f :

C k (E; G) → C k+1 (E; G) is defined, for all α ∈ C k (E; G)
F → F ′ is. If F * is the cellular sheaf obtained as F * = Hom F 2 (F ; F 2 ) from a cellular cosheaf F , then the complex (C k (E; F * ); d) k≥0 is the dual complex of (C k (E; F ); ∂) k≥0 . If E is the cubical subdivision of a simplicial complex K and F is a cubical cosheaf (resp. sheaf) we write (Ω k (K; F ); ∂) k≥0 instead of (C k (E; F ); ∂) k≥0 (resp. (Ω k (K; F ); d) k≥0 instead of (C k (E; F ); d) k≥0 ).
Definition 2.13. The homology of a cellular cosheaf F (resp. cohomology of a cellular sheaf) of a regular CW-complex E is the homology (resp. cohomology) of the associated chain complex (resp. cochain complex) from Definition 2.12. It is denoted by

(H k (E; F )) k≥0 (resp. (H k (E; F )) k≥0 ).
In the special case of definition 2.8 we write, for all p, q ≥ 0:

-H p,q (P ; F 2 ) instead of H q (K; F P p ); -H p,q (P ; F 2 ) instead of H q (K; F p P );

-H p,q (X; F 2 ) instead of H q (K; F X p ); -H p,q (X; F 2 ) instead of H q (K; F p X ).
Using the Universal Coefficient Theorem29 , the vector spaces H p,q (P ; F 2 ) and H p,q (P ; F 2 ) are dual of each other, and so are H p,q (X; F 2 ) and H p,q (X; F 2 ). We introduce the notations, for all p, q ≥ 0, i p,q : H p,q (X; F 2 ) → H p,q (P ; F 2 ) and i p,q : H p,q (P ; F 2 ) → H p,q (X; F 2 ) for the morphisms induced in homology and cohomology by the inclusions i p : F X p ⊂ F P p and there adjoint projections. By construction and the Universal Coefficient Theorem the morphism i p,q is the adjoint of i p,q . Definition 2.14 (Cup product). Let F be a cubical sheaf of algebras over F 2 on a simplicial complex K. The cochain complex (Ω k (K; F ); d) k≥0 is canonically endowed with the structure of graded algebra:

∪ : Ω k (K; F ) ⊗ F 2 Ω l (K; F ) -→ Ω k+l (K; F ) α ⊗ β -→ (σ p ≤ σ p+k+l ) → σ p ≤σ p+k ≤σ p+k+l α(σ p ; σ p+k )β(σ p+k σ p+k+l ) ,
where the product α(σ p ; σ p+k )β(σ p+k σ p+k+l ) is understood as the product of α(σ p ; σ p+k )

σ p ;σ p+k+l σ p ;σ p+k with β(σ p+k σ p+k+l ) σ p ;σ p+k+l σ p+k ;σ p+k+l in F (σ p ; σ p+k+l ). It satisfies the Leibniz rule: d(α ∪ β) = (dα) ∪ β + α ∪ dβ,
and therefore induces a well defined product in cohomology. In particular, (Ω k (K; F ); d; ∪) k≥0 is a graded differential algebra over F 2 .

Both the sheaves p∈N F p P and p∈N F p X are cubical sheaves of graded algebras for the wedge product. Moreover, the projection p∈N F p P → p∈N F p X is a morphism of sheaves of graded algebras. Therefore the collections of vector spaces (H p,q (P ; F 2 )) p,q∈N and (H p,q (X; F 2 )) p,q∈N both have the structure of bigraded algebras over F 2 and the cohomological "projection" p,q∈N i p,q is a morphism of bigraded algebras.

Construction of T-Hypersurfaces

Notations. Let P be a smooth, full dimensional, integral polytope of the n-dimensional vector space t * (R) endowed with the lattice t * (Z), and K be a primitive triangulation of P with dual hypersurface X. Definition 3.1. We denote by RP the topological space obtained as the quotient of t(F 2 ) × P by the equivalence relation:

(v; x) ∼ (v ′ ; x ′ ) if and only if x = x ′ and v -v ′ ∈ Sed(x) ,
where Sed(x) denotes the space {v ∈ t(Z) | α(v) = 0, ∀α ∈ T Q} ⊗ F 2 , Q being the only face of P containing x in its relative interior. This space is obtained by gluing together 2 n copies of P and is naturally a regular CW-complex. Using the moment map, this space is homeomorphic to Y P (R) the real locus of the toric variety associated with P . Therefore RP can be thought as a CW-structure on Y P (R). We denote by | • | : RP → P the projection onto P . It corresponds to the moment map. We denote by RK the lift of K to a subdivision of RP . It might not be a triangulation, however it is always a ∆-complex. To distinguish the cells of K from those of RK we will denote the former by the symbols σ p and the latter by the symbols σ p R . With this definition the projection | • | : RP → P is automatically a cellular map and for any simplex σ p ∈ K, the set of p-simplices of RK above σ p is canonically in bijection with the vector space:

t(F 2 ) Sed(σ p ) .
Thus we define the argument of a simplex σ p R ∈ RK, denoted by arg(σ p R ), to be the element of t(F 2 ) Sed(σ p ) to which σ p R corresponds. A dihomologic cube of RK is indexed by a pair of simplices σ p R ≤ σ q R and is projected by

|•| on the cube of K indexed by |σ p R | ≤ |σ q R |. Reciprocally, if σ p ≤ σ q is
a pair of simplices of K its set of lifting cubes is in bijection with the vector space:

t(F 2 ) Sed(σ p ; σ q ) . If the cube σ p R ≤ σ q R lifts the cube σ p R ≤ σ q R we define arg(σ p R ; σ q R )
as the element of t(F 2 ) Sed(σ p ; σ q ) to which it corresponds. One can show that arg(σ p R ; σ q R ) equals arg(σ q R ).

Definition 3.2. (Rule of Signs Cocycle) The cellular complex RK possesses a canonical 1cocycle ω RX ∈ Z 1 (RK; F 2 ) given, for all edges σ 1 R , by the following formula:

ω RX (σ 1 R ) := ω |σ 1 R | arg(σ 1 R ) .
The projection map |•| : RP → P being cellular, it induces both a morphism chain complexes and a morphism cochain complexes respectively denoted by

| • | : (C p (RK; F 2 ) → C p (K; F 2 )) p≥0 and by | • | * : (C p (K; F 2 ) → C p (RK; F 2 )) p≥0 for all primitive triangulations K of P . Definition 3.3 (T-Hypersurface). Let ε ∈ C 0 (K; F 2 ).
We may call such cochain a sign distribution on K. We denote by RX ε the subcomplex of the cubical subdivision of RK dual to the 1-cocycle d|ε| * + ω RX . That is to say, RX ε is the union of the closed cubes of RK indexed by the pairs of simplices

σ 1 R ≤ σ n R for which dε(|σ 1 R |) + ω RX (σ 1 R ) = 1.
We say that RX ε is the T-hypersurface constructed from the sign distribution ε. See Figure 4 for an example. We may say that the cohomology class

[ω RX ] ∈ H 1 (RP ; F 2 ) is the degree of RX ε .
Remark 3.4. Following the terminology of O. Viro, we may call the hypersurface RX ε a primitive patchwork when the primitive triangulation K is convex.

Proposition 3.5. RX ε is a sub-manifold of RP of codimension 1 that is Poincaré dual to the class [ω RX ] ∈ H 1 (RP ; F 2 ).
Proof. The first part of the statement is Proposition 4.11 p.26 of [START_REF] Brugallé | Combinatorial Patchworking: Back from Tropical Geometry[END_REF]. The second part is a consequence of both the definition of RX ε and of the Poincaré duality. See Theorem 67.1 p.394 of [START_REF] Munkres | Elements of Algebraic Topology[END_REF] for instance.

Example 3.6. In Figure 4 we give an example of T-curve in the compact torus P 1 (R) × P 1 (R) from a triangulation of the square of size 3. It is made of two connected components: one contractible, and one non-contractible whose homology class is (1; 1) in the canonical basis The image of RX ε under the projection | • | : RP → P does not depend on ε. It is always the dual hypersurface X. Consider a cube of K indexed by σ p ≤ σ q . We denote by Arg ε (σ p ; σ q ) the set of arguments arg(σ p R ; σ q R ) ∈ t(F 2 ) Sed(σ q ; σ q ) of the lifts σ p R ≤ σ q R of σ p ≤ σ q belonging to RX ε . This is a cubical cosheaf of sets.

([P 1 (R) × {0}], [{0} × P 1 (R)]). 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 (a)
Lemma 3.7. For all pairs σ p ≤ σ q of simplices of K the set Arg ε (σ p ; σ q ) is the complementary sub-set in V = t(F 2 ) Sed(σ q ; σ q ) of an affine sub-space parallel to {v ∈ V | α(v) = 0, ∀α ∈ T σ p (F 2 )}, so in particular of codimension p.

Proof. The set Arg ε (σ p ; σ q ) is the set of the vectors v ∈ V for which there exists σ 1 ⊂ σ p such that: dε(σ 1 ) + ω(σ 1 )(v) = 1.

We can see a vector v ∈ V as a closed 1-cocycle on σ p using the following morphism:

z : V -→ Z 1 (σ p ; F 2 ) v -→ [σ 1 → ω(σ 1 )(v)] .
From that point of view Arg ε (σ p ; σ q ) is the set {v ∈ V | z(v) ̸ = dε| σ p } and it is the complementary sub-set in V of the fiber z -1 (dε| σ p ). This fibre is an affine space parallel to the kernel of z. Since K is primitive, z is onto and its kernel has codimension p. Furthermore, the family of linear forms {v → z(v)(σ 1 ) :

σ 1 ≤ σ p } spans T σ p (F 2 ) so the kernel of z is precisely {v ∈ V | α(v) = 0, ∀α ∈ T σ p (F 2 )}.
Remark 3.8. Let σ 1 be an an edge of K. The sets Arg ε (σ 1 ; σ n ) are all identical for all nsimplices σ n ≥ σ 1 . A. Renaudineau and K. Shaw denoted a member of this collection by E and called it the real phase structure.

Definition 3.9 (Sign Sheaves and Cosheaves). We set the notations of two cubical cosheaves and two cubical sheaves on K:

1. K RP is the cosheaf of group algebras (σ p ; σ q ) → F 2 F P 1 (σ p ; σ q ) (we will denote by x v the generator associated with v ∈ F P 1 (σ p ; σ q )) and its dual, the sheaf O RP associates the F 2 -algebra of functions from F P 1 (σ p ; σ q ) to F 2 to the pair (σ p ; σ q ).

2. Likewise we set K RXε to be the sub cubical cosheaf of K RP that associates to (σ p ; σ q ) the sub-space of F 2 F P 1 (σ p ; σ q ) freely generated by the elements of Arg ε (σ p ; σ q ). Its dual, O RXε , is the quotient of O RP that associates to (σ p ; σ q ) the algebra of functions from Arg ε (σ p ; σ q ) to F 2 . 3. We denote by i * : K RXε → K RP the inclusion and by i * : O RP → O RXε the adjoint projection. The latter morphism is a morphism of sheaves of algebras.

With this definition we can recall Proposition 3.17 of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] and translate it for the dual sheaves.

Proposition 3.10 (Folding Isomorphisms). There are chain complexes isomorphisms Φ RP and Φ RXε that make the following diagram commute:

Ω q (K; K RP ) q≥0 Ω q (RK; F 2 ) q≥0 Ω q (K; K RXε ) q≥0 C q (RX ε ; F 2 ) q≥0 ∼ = Φ RP ∼ = Φ RXε i *
As a consequence the dual isomorphisms make the dual diagram commute:

Ω q (K; O RP ) q≥0 Ω q (RK; F 2 ) q≥0 Ω q (K; O RXε ) q≥0 C q (RX ε ; F 2 ) q≥0 i * ∼ = Φ * RP ∼ = Φ *

RXε

Proof. The first part of the statement is Proposition 3.17 of p.19 in [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF]. The second part is a consequence of the duality. We can however remind ourselves of the construction of Φ RP (Φ RXε is its restriction). If σ p ≤ σ q is a cube of K and v ∈ t(F 2 ) Sed(σ q ) let us write (v; σ p ; σ q ) for the corresponding lift in RK. The isomorphism Φ RP sends the chain x v ⊗ (σ p ; σ q ) to the chain (v; σ p ; σ q ).

Renaudineau-Shaw Spectral Sequence

We recall the construction of the filtration given by A. Renaudineau and K. Shaw in [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] and describe the dual filtration in cohomology.

The Augmentation Filtration in Group Algebras. We denote respectively by Vect. f F 2 , Hopf f F 2 , and gr.Alg. f F 2 the categories of finite dimensional vector spaces over F 2 , of finite dimensional Hopf algebras over F 2 , and of finite dimensional graded algebras over F 2 . Definition 4.1. We denote by F 2 [-] : Vect. f F 2 → Hopf f F 2 the functor that associates to a finite dimensional vector space its group ring over F 2 . For all finite dimensional vector spaces V over F 2 , we denote by x v ∈ F 2 [V ] the generator associated with the vector v ∈ V . The generator x 0 is the unit of F 2 [V ]. Hence it equals 1. We use the notation m V for the augmentation ideal of F 2 [V ]. This is the kernel of the augmentation of F 2 [V ]: the morphism of algebras aug V : F 2 [V ] → F 2 that sends every generator x v to 1. Note that this is the only algebra morphism F 2 [V ] → F 2 . This ideal is known 30 to be generated by the elements of the form 1 + x v for all v ∈ V . We denote by gr : Hopf f F 2 → gr.Alg. f F 2 the functor that associates to a Hopf algebra A the graded algebra builded from the filtration of iterated powers of its augmentation ideal. For instance if V is a finite dimensional vector space over F 2 we have:

gr F 2 [V ] = k≥0 m k V m k+1 V .
Moreover, for all k ∈ N, we denote the functor V → m k V by m k : Vect. f F 2 → Vect. f F 2 . These functors are linked by natural inclusions m l → m k for all l ≥ k. where W is a sub-vector space of dimension k of V .

Proposition 4.3 (F. Bihan, M. Franz, C. McCrory, and J. van Hamel, [BFMvH06] Proposition 6.3 p.8). We denote by : Vect. f F 2 → gr.Alg. f F 2 the exterior algebra functor. There is a natural isomorphism η :

→ gr F 2 [-] defined for all finite dimensional F 2 -vector spaces V , and all k ∈ N, by the formula:

η k V : k V -→ m k V m k+1 V v 1 ∧ ... ∧ v k -→ k i=1 (1 + x v i ) mod m k+1 V .
This description is a special case of D. Quillen's description of graded rings associated with the augmentation filtration of a group algebras31 . The proof of the special case of interest here is recalled in [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] under Proposition 4.3 on p.21.

Proposition 4.4. Let V be a finite dimensional vector space over F 2 . For all v ∈ V , the graded endomorphism of V ∼ = gr F 2 [V ] induced by the multiplication by x v is the identity.

Proof. Let v be a vector of V and v 1 , ..., v k ∈ V be a linearly independent family of V , we have:

k i=1 (1 + x v i ) = x v k i=1 (1 + x v i ) + (1 + x v ) k i=1 (1 + x v i ). Hence k i=1 (1 + x v i ) and x v k i=1 (1 + x v i ) correspond to the same element in m k V m k+1 V .
Proposition 4.5. Let V be a finite dimensional vector space over F 2 . For all linear sub-spaces W ⊂ V and all k ∈ N, the intersection of the sub-algebra

F 2 [W ] of F 2 [V ] with m k V is the ideal m k W .
Proof. This is a consequence of the functoriality of F 2 [-] and the natural transformation η described in Proposition 4.3. However, we can also give a direct proof here. Using lemma 4.1 p.21 of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] we know that a generating set of m k V is given by the elements u∈U x u where U is a k-linear sub-space of V . Now let H be a linear hyperplane of V and

P = N i=1 u∈U i x u ∈ F 2 [H] ∩ m k V .
We want to write this element as a sum

N i=1 u∈U ′ i x u where U ′ i is a linear sub-space of H. Since P belongs to F 2 [H] we have: N i=1 u∈U i \H x u = 0.
Let v be the generator of a supplementary line of H. We define U ′ i as follows:

1. If U i ⊂ H, we set U ′ i := U i ; 2. If U i ̸ ⊂ H and (U i ∩ H) ̸ = v + (U i \ H) , we set U ′ i := (U i ∩ H) ∪ v + (U i \ H) ; 3. Otherwise we set U ′ i := ∅.
Note that in each case U ′ i ⊂ H and is ether a k-dimensional vector space or the empty set. Now we have:

N i=1 u∈U i x u = N i=1 u∈U i ∩H x u + N i=1 u∈U i \H x u = N i=1 u∈U i ∩H x u + x v N i=1 u∈U i \H x u = N i=1   u∈U i ∩H x u + u+v∈U i \H x u   = N i=1 u∈U ′ i x u . Thus F 2 [H] ∩ m k V = m k H .
Let W be a sub-space of codimension l ≥ 1 and H 1 , ..., H l be hyperplanes that intersect along W . We have:

F 2 [W ] ∩ m k V = F 2 [∩ l i=1 H i ] ∩ F 2 [∩ l i=2 H i ] ∩ ... ∩ F 2 [H l ] ∩ m k V = F 2 [∩ l i=1 H i ] ∩ F 2 [∩ l i=2 H i ] ∩ ... ∩ F 2 [H l-1 ∩ H l ] ∩ m k H l . . . = F 2 [∩ l i=1 H i ] ∩ m k ∩ l i=2 H i = m k W , since ∩ l i=k H i is a hyperplane of ∩ l i=k+1 H i for all 1 ≤ k ≤ l.
In the category of finite dimensional Hopf algebras there is a duality contravariant functor. We denote it by (-) * : Hopf f

F 2 op → Hopf f F 2 .
If we only remember the vector space structure, it corresponds to the classical functor (-) * = Hom F 2 (-; F 2 ). If A is a finite dimensional Hopf algebra, the product of A * is the adjoint of the coproduct of A, the unit of A * is the adjoint of the counit of A and vice-versa. Definition 4.6. We denote by O the functor

F 2 [-] * : Vect. f F 2 op → Hopf f F 2 .
The functor O is naturally isomorphic to the functor of F 2 -valued function algebras V → {f : V → F 2 } with pointwise addition and multiplication. We will not describe the coalgebra structure as we will not need it further. The natural isomorphism is realised by means of a duality pairing. For all finite dimensional vector spaces V , all functions f ∈ O(V ) and all elements P = v∈V p v x v ∈ F 2 [V ], we define such pairing by:

⟨ f ; P ⟩ := v∈V f (v)p v .
Remark 4.7. We justify the notation O(V ) by the known fact that, for all finite dimensional vector spaces V over a finite field F, every function f : V → F is polynomial. It is only a matter of expressing the indicatrix functions as polynomials in a set of coordinates of V .

(k) : Vect. f F 2 op → Vect. f
F 2 is given, for all k ∈ N and all finite dimensional vector spaces V over F 2 , by the formula32 :

O (k) (V ) := im F 2 [V ] / m k+1 * → O(V ) .
Furthermore, they are naturally included in one another O (k) → O (l) , for all l ≥ k. We denote by gr O : Vect. f

F 2 op → gr.Vect. f F 2 the functor k≥0 O (k) / O (k-1) .
By construction and Proposition 4.3 we have:

Proposition 4.9. The functor gr O is naturally isomorphic to (-) * through η * .

Since we did not define a product on gr O, we should emphasise that in Proposition 4.9 the two functors gr O and (-) * are naturally isomorphic as functors to the category of graded vector spaces. However, Proposition 4.11 will ensure that we can endow gr O with a product compatible with η * . Lemma 4.10. For all functions u : {1, ..., l} → {1, ..., k}, the number:

β∈F k 2 l j=1 β u(j) ∈ F 2 , equals 1 if and only if u is surjective.
Proof. If u is onto the only non-zero term in the sum is the one for which β i = 1 for all i. If on the contrary u is not surjective it misses an index i 0 and we have:

β∈F k 2 l j=1 β u(j) = β∈F k 2 β i 0 =0 l j=1 β u(j) + β∈F k 2 β i 0 =1 l j=1 β u(j) = 0 ,
since the two terms are equals.

Proposition 4.11. The composition

O (k) ⊗ O (l) → O ⊗ O → O
where the first natural transformation is given by the tensor product of the inclusions and the second is given by the product of O takes values in O (k+l) . In other words O (k) O (l) ⊂ O (k+l) .

Proof. Let V be a finite dimensional vector space over F 2 . We need to show that for all k, l ∈ N, all f ∈ O (k) (V ), and all g ∈ O (l) (V ), the product f g belongs to O (k+l) (V ). To do so we will show that a function is an element of O (k) (V ) if and only if it can be represented by a polynomial of degree at most k (c.f. Remark 4.7). The functions of O (k) (V ) are those which vanish against elements of m k+1 V . Following lemma 4.1 p.21 of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF], the vector space m k+1 V is generated by the elements v∈W x v for all linear sub-spaces W of V of dimension (k + 1). Such an element can be written as k+1 i=1 (1 + x w i ) where {w 1 , ..., w k+1 } is a basis of W . Let us fix a set of linear coordinates x 1 , ..., x n of V , i.e. a basis of V * . Because of Fermat's little theorem33 , a function f ∈ O(V ) can be uniquely written as follows:

f = α∈{0,1} n f α x α ,
in the usual multi-index notations: x α := n i=1 x α i i and |α| := n i=1 α i . For all integers k ≥ 1, let us denote by [k] the integer interval {1, ..., k}. Let α ∈ F n 2 with |α| = l and W = ⟨w 1 , ..., w k ⟩ be a k-dimensional sub-space of V . We have:

x α ; k i=1 (1 + x w i ) = w∈W ⟨x α ; x w ⟩ = w∈W n j=1 x j (w) α j = β∈F k 2 n j=1 x j k i=1 β i w i α j = β∈F k 2 u:[n]→[k] n j=1 β α j u(j) n j=1
x j (w u(j) )

α j
.

Up to reordering we can assume that α i = 1 if and only if 1 ≤ i ≤ l, thus:

β∈F k 2 u:[n]→[k] n j=1 β α j u(j) n j=1
x j (w u(j) )

α j = β∈F k 2 u:[l]→[k] l j=1 β u(j) l j=1 x j (w u(j) ) = u:[l]→[k]   β∈F k 2 l j=1 β u(j)   l j=1
x j (w u(j) ) .

By Lemma 4.10, the number β∈F k 2 l j=1 β u(j) equals 1 if and only if u is onto. Therefore, the number:

x α ; k i=1 (1 + x w i ) ,
vanishes whenever k is bigger than l. Thus the polynomial functions of degree at most k belong to O (k) (V ). If k equals l, and S k denotes the k-th symmetric group, the number:

x α ; k i=1 (1 + x w i ) = u∈S k l j=1 x j (w u(j) ) ,
is the minor of the matrix of the coordinates of w 1 , ..., w k in the dual basis of x 1 , ..., x n with the row set selected by the multi-index α. As a consequence we can find, for all polynomials of degree k, a sub-space W of dimension k such that the evaluation of f , as a linear form, against the generator of m k V associated with W is 1. It implies that {f : deg f = k} is included in

O (k) (V ) \ O (k-1) (V )
and that the two partitions of O(V ), by the degree on one hand, and by the sets O (k) (V ) \ O (k-1) (V ) on the other hand, coincide.

As a direct consequence of Proposition 4.11 we have:

Proposition 4.12. The functor O together with the filtration (O (k) ) k∈N takes values in the category of filtered finite dimensional algebras over F 2 , and therefore gr O takes values in the category of graded algebras.

Moreover:

Proposition 4.13. The natural isomorphism η * : gr O → (-) * is compatible with the products.

Proof. Let V be a finite dimensional vector space over F 2 and let k ∈ N. Let us consider f ∈ O (k) (V ) and express it in a system of linear coordinates x 1 , ..., x n :

f = |α|≤k f α x α = I⊂{1,..,n} |I|≤k f I i∈I x i .
Using Lemma 4.10, we find that, for all v 1 ∧ ...

∧ v k ∈ k V : f mod O (k-1) (V ) ; η k V (v 1 ∧ ... ∧ v k ) = |I|=k f I i∈I x i ; k j=1 (1 + x v j ) = |I|=k f I u∈S k i=1 x I(i) (v u(i) ) = |I|=k f I i∈I x i n i=1 v i ,
where I(i) is the i-th element of I. The inverse of (η k V ) * sends a wedge product of linear forms α 1 ∧ ... ∧ α k to the polynomial function α 1 ...α k modulo O (k-1) (V ). It respects the product structures.

Definition 4.14. Let V be a finite dimensional vector space over F 2 . For all f ∈ O(V ) and all P = v∈V p v x v ∈ F 2 [V ] we denote by f • P the element of F 2 [V ] defined by the formula:

f • P := v∈V f (v)p v x v .
Proposition 4.15. Let V be a finite dimensional vector space over F 2 . For all f ∈ O (l) (V ) and all P ∈ m k+l V , f • P belongs to m k V . Moreover, the induced product between the graded pieces:

l V * ⊗ k+l V → k V ,
is the contraction.

Proof. Let a ∈ O (1) (V ) be an affine function and denote its linear part by da. A straightforward computation yields for all P, Q ∈ F 2 [V ]:

a • (P Q) = (a • P )Q + P (da • Q) .
Using this formula we can show, by induction, that a•m k+1 V is included in m k V for all k ≥ 0. From the definition (f g)

• P equals f • (g • P ) for all f, g ∈ O(V ) and all P ∈ F 2 [V ]. Consequently, (a 1 ...a l ) • m k+l V is included in m k
V for all a 1 , ..., a l ∈ O (1) (V ). Finally, we find the following inclusions:

O (l) (V ) • m k+l V ⊂ l i=0 m k+l-i V ⊂ m k V .
The duality pairing ⟨f ; P ⟩ of f ∈ O(V ) with P ∈ F 2 [V ] can be written as aug V (f • P ). Hence, the operation P → f • P is adjoint to g → gf . This remains true in the graded algebras and it is precisely the definition of the contraction.

Lemma 4.16. Let V be a finite dimensional vector space over F 2 and A be an affine sub-space of V of codimension l. Let χ A denote the indicatrix function of A. We have the following exact sequence:

0 ⟨ x v : v ∈ V \ A ⟩ F 2 [V ] ⟨ x v : v ∈ A ⟩ 0 . χ A •- (1)
Moreover, if ω denotes the generator of

l {α | α(v) = 0, ∀v ∈ T A}, K denotes ⟨ x v : v ∈ V \A ⟩,
and o is a point of A, then for all k ∈ N, we have the following diagram with exact rows and columns:

0 0 0 0 K ∩ m k V K ∩ m k+1 V k V k-l T A 0 0 K ∩ m k V m k V m k T A 0 0 K ∩ m k+1 V m k+1 V m k+1 T A 0 0 0 0 ω•- x o (χ A •-) x o (χ A •-) (2) Proof. The space F 2 [V ] is the direct sum of ⟨ x v : v ∈ A ⟩ and ⟨ x v : v ∈ V \ A ⟩. A direct computation yields that the operation P → χ A • P is the projection onto ⟨ x v : v ∈ A ⟩ parallel to ⟨ x v : v ∈ V \ A ⟩.
Therefore we have the exact sequence (1):

0 ⟨ x v : v ∈ V \ A ⟩ F 2 [V ] ⟨ x v : v ∈ A ⟩ 0 χ A •- .
Since A has codimension l, it is the intersection of l affine hyperplanes. The indicatrix functions of affine hyperplanes have degree 1 hence χ A has degree l. Following Proposition 4.15, the linear map P → χ A • P has degree (-l) as a morphism of filtered vector spaces. Moreover, because of Proposition 4.4, choosing a point o ∈ A yields an isomorphism of filtered vector spaces

P ∈ F 2 [T A] → x o P ∈ ⟨ x v : v ∈ A ⟩. Therefore, P → x o (χ A • P ) is a surjective filtered morphism of degree (-l) from F 2 [V ] onto F 2 [T A].
Consequently, for all k ∈ N, we obtain, by partial use of the Snake Lemma34 , the following commutative diagram with exact rows and columns:

0 0 0 k V k-l T A C k 0 m k V m k-l T A C (k) 0 m k+1 V m k-l+1 T A C (k+1) 0 0 0 f k x o (χ A •-) x o (χ A •-)
Where f k is induced by x o (χ A • -) between the k-th graded pieces and the right hand side column is made of the cokernels of the associated morphisms. Using Proposition 4.4 and Proposition 4.15 we find that the morphism f k is the contraction against ω. It is a surjective morphism and C k vanishes. Since all the C k 's vanish the C (k) 's must vanish as well. Using now the Snake Lemma in its complete form we find the commutative diagram (2).

Proposition 4.17. Let V be a finite dimensional vector space over F 2 and A ⊂ V be an affine sub-space of codimension l. For all families H 1 , ..., H l ⊂ V of affine hyperplanes that intersect along A and all k ∈ N, we have:

⟨ x v : v ∈ V \ A ⟩ ∩ m k V = l i=1 ⟨ x v : v ∈ V \ H i ⟩ ∩ m k V .
Proof. Let us denote by K the sub-space

⟨ x v : v ∈ V \ A ⟩ of F 2 [V ], also equal to l i=1 ⟨ x v : v ∈ V \ H i ⟩.
It is endowed with two decreasing filtrations:

K (k) := ⟨ x v : v ∈ V \ A ⟩ ∩ m k V ,
and:

K ′ (k) := l i=1 ⟨ x v : v ∈ V \ H i ⟩ ∩ m k V .
We need to show that they coincide. Since K ′ (k) is included in K (k) for all k ∈ N, we can only show that the induced morphisms between the graded pieces are surjective. Because of Proposition 4.4 and Proposition 4.5,

K ′ (k) is equal to l i=1 x o i m k T H i where, for all 1 ≤ i ≤ l, o i is a point of V \ H i and T H i is the direction of H i . It follows that the image of K ′ (k) in k V is: l i=1 k T H i .
As a consequence of the definition of the other filtration (K (i) ) i≥0 , its associated graded vector space is embedded in the exterior algebra V . Moreover, if ω is the generator of l ( V / T A ) * and k is a non-negative integer, we have, from Diagram (2) of Lemma 4.16, the following exact sequence:

0 K k k V k-l T A 0 c k ω•- , (3)
and the commutative diagram:

K ′ k k V K k a k b k c k Where a k is induced from the inclusions (K ′ (k) ⊂ K (k) ) k∈N , b k is induced from (K ′ (k) ⊂ m k V ) k∈N , and c k is induced from (K (k) ⊂ m k V ) k∈N .
The morphism c k is injective and using Proposition 2.11 with the Exact Sequence (3) imply that c k and b k have the same image, so that a k is onto. It imlplies that all the inclusions K ′ (k) ⊂ K (k) , for all k ∈ N are onto, hence equalities.

The Induced Filtrations.

Definition 4.18. In Definition 3.9 the cubical cosheaf K RP and the cubical sheaf O RP were given by the formulae K RP = F 2 [F P 1 ] and O RP = O(F P 1 ). The functoriality of the filtrations of F 2 [-] and O ensures that these cosheaf and sheaf are respectively endowed with a decreasing filtration:

0 ⊂ K RP (n) ⊂ ... ⊂ K RP (0) = K RP
, and an increasing filtration:

0 ⊂ O (0) RP ⊂ ... ⊂ O (n) RP =
O RP , and that the associated graded pieces satisfy, for all k ∈ N:

0 -→ K RP (k+1) -→ K RP (k) -→ bv P k F P k -→ 0 , (4)
and:

0 -→ O (k-1) RP -→ O (k) RP -→ bv k P F k P -→ 0 . ( 5
)
The right hand side projections bv P k and bv k P are given by the projections onto the graded pieces composed with the natural isomorphisms η and (η * ) -1 . We denote by: -E r p,q (RP ) p,q,r≥0 the spectral sequence converging toward H q (RP ; F 2 ) q≥0 that arrises from the bounded filtration of K RP ; -E p,q r (RP ) p,q,r≥0 the dual spectral sequence converging toward H q (RP ; F 2 ) q≥0 that arrises from the bounded filtration of O RP .

The spectral sequence E p,q r (RP ) p,q,r≥0 is a spectral ring for O RP is a sheaf of filtered algebras35 . These filtrations can be transported on RX ε : Definition 4.19. The filtration of K RP from Definition 4.18 induces a filtration of the subcosheaf K RXε ⊂ K RP by intersection. Dually, the filtration of O RP induces a filtration of O RXε by projection. With these definitions the inclusion i * : K RXε → K RP and the projection i * : O RP → O RXε are morphisms of filtered cosheaves and sheaves respectively. Lemma 1.7 ensures that these filtrations are dual of one another. We denote by: E r p,q (RX ε ) p,q,r≥0 and E p,q r (RX ε ) p,q,r≥0 , the spectral sequences respectively associated with the filtrations of K RXε and of O RXε . The sheaves and cosheaves morphisms i * and i * induce spectral sequences morphisms:

i r p,q : E r p,q (RX ε ) → E r p,q (RP ) and i p,q r : E p,q r (RP ) → E p,q r (RX ε ) .

The multiplicative property of the filtration of O RP is passed onto the filtration of O RXε since i * : O RP → O RXε is a morphism of sheaves of algebras and therefore E p,q r (RX ε ) p,q,r≥0 is also a spectral ring. Furthermore, the collection (i p,q r ) p,q,r≥0 is a morphism of spectral rings. The main theorem of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] rests on the computation of the first page of the spectral sequence E r p,q (RX ε ) p,q,r≥0 and more specifically the graded pieces of K RXε . However, we gave a different definition of the filtration of K RXε . Before recalling the results of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF] about the spectral sequence E r p,q (RX ε ) p,q,r≥0 we should check that the two definitions coincide. Let us recall their constriction.

22). Let us denote by (K RXε

(k) ) ′ k≥0 the decreasing filtration of K RXε constructed as follows: 1. Let σ 1 ≤ σ q be a pair of simplices of K. The set Arg ε (σ 1 ; σ q ) is a hyperplane of the vector space F P 1 (σ 1 ; σ q ) = t(F 2 ) Sed(σ 1 ; σ q ) (c.f. Lemma 3.7). Let us denote its vectorial direction by T A and chose an origin o ∈ Arg ε (σ 1 ; σ q ). Let k ∈ N, and (K RXε (k) ) ′ (σ 1 ; σ q ) denote the image of the k-th power augmentation ideal m k T A by the isomorphism F 2 [T A] → K RXε (σ 1 ; σ q ) obtained from the multiplication by x o . 2. Let σ p ≤ σ q be a pair of simplices of K and k ∈ N. The sub-space (K RXε (k) ) ′ (σ p ; σ q ) of K RXε (σ p ; σ q ) is obtained as the sum:

(K RXε (k) ) ′ (σ p ; σ q ) := σ 1 ≤σ p (K RXε (k) ) ′ (σ 1 ; σ q ) .
Proposition 4.21. Let σ p ≤ σ q be a pair of simplices of K. For all integers k ≥ 0:

(K RXε (k) ) ′ (σ p ; σ q ) = K RXε (k) (σ p ; σ q ) .
Proof. Let k ≥ 0 be an integer. From Definition 4.19, we have:

K RXε (k) (σ p ; σ q ) = x v : v ∈ Arg ε (σ p ; σ q ) ∩ m k V ,
where V is t(F 2 ) Sed(σ p ; σ q ) . Lemma 3.7 ensures that Arg ε (σ p ; σ q ) is the complementary subset of an affine sub-space of V . Let σ 0 be a vertex of σ p . We can apply Proposition 4.17 to find that:

x v : v ∈ Arg ε (σ p ; σ q ) ∩ m k V = σ 0 ⊂σ 1 ⊂σ p x v : v ∈ Arg ε (σ 1 ; σ q ) ∩ m k V .
Indeed, the sets V \ Arg ε (σ 1 ; σ q ), for all edges σ 0 ⊂ σ 1 ⊂ σ p , are hyperplanes intersecting along V \ Arg ε (σ p ; σ q ). We can add the remaining x v : v ∈ Arg ε (σ 1 ; σ q ) ∩ m k V on both sides to find that:

x

v : v ∈ Arg ε (σ p ; σ q ) ∩ m k V = σ 1 ⊂σ p x v : v ∈ Arg ε (σ 1 ; σ q ) ∩ m k V .
Because of Proposition 4.4 and Proposition 4.5, the intersection of the sub-space

x v : v ∈ Arg ε (σ 1 ; σ q ) with m k V corresponds to (K RXε (k) ) ′ (σ 1 ; σ q ).
Using the second construction step of Definition 4.20 we finally find that the two filtrations coincide.

The following proposition describes the graded pieces of the filtration of K RXε and is mainly equivalent to Lemma 4.8 and Proposition 4.10 of [START_REF] Renaudineau | Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit[END_REF]. However in this article, the Borel-Viro morphisms (bv X k ) k∈N were not introduced as we do here since their construction relied on a different definition of the filtration, i.e. Definition 4.20.

Proposition 4.22. For all integers k ≥ 0, the Borel-Viro morphism bv P k of Definition 4.18 sends K RXε (k) onto F X k . We denote its restriction-corestriction by bv X k . As a consequence we have the following commutative diagram with exact rows:

0 K RP (k+1) K RP (k) F P k 0 0 K RXε (k+1) K RXε (k) F X k 0 0 0 0 bv P k i * bv X k i * i k (6)
Proof. Let σ p ≤ σ q be a pair of simplices of the triangulation K and k ∈ N. From Diagram (2) of Lemma 4.16 and Proposition 2.11 we derive the following commutative diagram with exact rows and columns:

0 0 0 0 F X k (σ p ; σ q ) F P k (σ p ; σ q ) k-p T A 0 0 K RXε (k) (σ p ; σ q ) K RP (k) (σ p ; σ q ) m k-p T A 0 0 K RXε (k+1) (σ p ; σ q ) K RP (k+1) (σ p ; σ q ) m k+1-p T A 0 0 0 0 i k ω•- i * bv X k x o (χ A •-) bv P k bv T A k i * x o (χ A •-)
where A is the codimension p affine sub-space of F P 1 (σ p ; σ q ) complementary to Arg ε (σ p ; σ q ), o is a point of A, and ω is the generator of

p F P 1 (σ p ; σ q ) / T A * . Remark 4.23. If x o k i=1 (1+x v i ) belongs to K RP (k) (σ p ; σ q ), its image under bv P k (σ p ; σ q ) is k i=1 v i .
Proposition 4.24. The Borel-Viro morphism bv k P of Definition 4.18 composed with i k :

F k P → F k X can be factored by i * : O (k) RP → O (k)
RXε and we denote this factorisation by bv k X . As a consequence we have the following commutative diagram with exact rows and columns:

0 O (k-1) RP O (k) RP F k P 0 0 O (k-1) RXε O (k) RXε F k X 0 0 0 0 i * bv k P i * i k bv k X (7)
Proof. Let σ p ≤ σ q be a pair of simplices of the triangulation K and k ∈ N. By dualising Diagram (2) of Lemma 4.16 and using Proposition 2.11 we derive the following commutative diagram with exact rows and columns:

0 0 0 0 k-p (T A) * F k P (σ p ; σ q ) F k X (σ p ; σ q ) 0 0 O (k-p) (T A) O (k) RP (σ p ; σ q ) O (k) RXε (σ p ; σ q ) 0 0 O (k-1-p) (T A) O (k-1) RP (σ p ; σ q ) O (k-1) RXε (σ p ; σ q ) 0 0 0 0 ω∧- i k ext A bv k-p T A i * bv k P bv k X ext A i *
where A is the codimension p affine sub-space of F P 1 (σ p ; σ q ) defined as the complementary sub-set of Arg ε (σ p ; σ q ), ext A is the composition of the translation isomorphism:

f ∈ O(T A) → [x → f (x + o)] ∈ O(A),
for a choice of o ∈ A, with the extension by 0 to a map of O F P 1 (σ p ; σ q ) , and ω is the generator of the dual of p F P 1 (σ p ; σ q ) / T A .

Properties of the Spectral Sequences. The two following theorems describes the already known properties of the Renaudineau-Shaw spectral sequences.

Theorem 4.25 (A. Borel and A. Haefliger, [BH61], §5.15 p.496). The spectral sequences E r p,q (RP ) p,q,r≥0 and E p,q r (RP ) p,q,r≥0 degenerate at the first page (i.e. ∂ r and d r vanish for all integers r ≥ 1). Moreover, the graded algebra H q (RP ; F 2 ); ∪ q≥0 is isomorphic to H q,q (P ; F 2 ); ∪ q≥0 . Theorem 4.26 (Structure of the Spectral Sequences of RX ε , A. Renaudineau and K. Shaw, [RS23] Lemma 6.3 p.26 and Proposition 4.12 p.24). The first page E 1 p,q (RX ε ) p,q≥0 is isomorphic to the tropical homology H p,q (X; F 2 ) p,q≥0 . Moreover for all integers r ≥ 2, the only possibly non-trivial groups of the r-th page E r p,q (RX ε ) p,q≥0 are located on the line segments {p = q and 0 ≤ q ≤ n -1} and {p + q = n -1 and 0 ≤ q ≤ n -1}. Figure 5 depicts the shape of such a spectral sequence. Hence, the only possibly non-trivial boundary operators (∂ r p,q ) p,q≥0 are:

∂ r n-r 2 , n-r 2 and ∂ r n-r 2 -1, n+r 2 ,
when r is congruent to n modulo 2. By duality the same holds for the dual page E p,q r (RX ε ) p,q≥0 . However, this time the possibly non-trivial differentials are:

d n+r 2 , n-r 2 -1 r and d n+r 2 -1, n+r 2 -1 r
, under the same hypothesis r = n (mod 2).

E 3,3 1 0 0 0 E 2,2 1 0 E 0,3 1 E 1,2 1 E 2,1 1 E 3,0 1 0 E 1,1 1 0 0 0 E 0,0 1 (a)
The first page.

E 3,3 2 0 0 0 E 2,2 2 0 E 0,3 2 E 1,2 2 E 2,1 2 E 3,0 2 0 E 1,1 2 0 0 0 E 0,0 2 (b)
The second page.

E 3,3 3 0 0 0 E 2,2 3 0 E 0,3 3 E 1,2 3 E 2,1 3 E 3,0 3 0 E 1,1 3 0 0 0 E 0,0 3 (c)
The third page. Following Theorem 4.26 we will say that a page of the spectral sequences of RX ε is irrelevant if its differentials or boundary operators are all trivial for having either a trivial source or a trivial target. The other ones, which we call the relevant pages are those for which 0 ≤ r ≤ 1, or 2 ≤ r ≤ n -1 and r = n (mod 2). Theorem 4.27 (Symmetry). For all r ≥ 1, the E r -pages of the Renaudineau-Shaw spectral sequence computing the cohomology of RX ε satisfy the Poincaré duality. That is to say:

1. The vector space E p,q r (RX ε ) vanishes whenever p > n -1 or q > n -1;

2. The vector space E n-1,n-1 r (RX ε ) has dimension 1;

3. The bilinear pairing ∪ :

E p,q r (RX ε ) ⊗ E n-1-p,n-1-q r (RX ε ) → E n-1,n-1 r (RX ε ) is non- degenerate.
In particular, E p,q r (RX ε ) is isomorphic to E n-1-p,n-1-q r (RX ε ) and d p,q r has the same rank and kernel dimension as d n-1-p+r,n-2-q r . Proof. By construction the first page of the spectral sequence (E p,q 1 (RX ε ) p,q≥0 is the cohomology of the sheaf of graded algebras associated with O RXε . Following Proposition 4.24, this sheaf is isomorphic to the sheaf of graded algebras n-1 k=1 F k X . On that account E p,q 1 (RX ε ) is isomorphic, for all p, q ∈ N, to the tropical cohomology group H p,q (X; F 2 ). Furthermore, this isomorphism respects the cup product. Using Theorem 5.3 p.16 of [START_REF] Jell | Lefschetz (1,1)-theorem in tropical geometry[END_REF], this page satisfies the Poincaré duality and the theorem follows from Lemma 1.5.

Corollary 4.28. If RX ε is even dimensional, i.e. P has odd dimension, then the total Betti number of RX ε satisfies:

n-1 i=1 dim H i (RX ε ; F 2 ) = n-1 p,q=0 dim H p,q (X; F 2 ) (mod 4) .
Proof. By the Universal Coefficient Theorem36 and duality the statement of the corollary is equivalent to:

n-1 i=1 dim H i (RX ε ; F 2 ) = n-1 p,q=0 dim H p,q (X; F 2 ) (mod 4) .
Using the convergence of the spectral sequence (E p,q r (RX ε )) p,q,r≥0 and Theorem 4.26 we find that:

n-1 i=1 dim H i (RX ε ; F 2 ) = n-1 p,q=0 dim E p,q 1 (RX ε ) - r≥1 n-1 p,q=0 2 rk d p,q r = n-1 p,q=0 dim E p,q 1 (RX ε ) - p+q=n-1 0≤q≤n-2 2 rk d p,q 1 U - r≥2 r=n(mod 2) 2(rk d n+r 2 , n-r 2 -1 r + rk d n+r 2 -1, n+r 2 -1 r ) V .
From Theorem 4.27 we know that d 1 has the same rank as the differential d q+1,p-1 1 . Thus U is also divisible by 4 whenever n is odd. Finally we deduce Corollary 4.28 by remembering that E p,q 1 (RX ε ) is isomorphic to H p,q (X; F 2 ).

Remark 4.29. It was pointed out to me by I. Itenberg in a conference that Corollary 4.28 could already be deduced from the fact that the Euler characteristic χ(RX ε ) is equal to the signature of a non-singular hypersurface CX of the complex locus of the toric variety associated with P and arising from a section the ample line bundle associated with the polytope37 . Indeed, because RX ε is of even dimension 2k and satisfies the Poincaré duality with F 2 -coefficients its total Betti number is congruent to (-1) k χ(RX ε ) modulo 4. From Lemma 2.2 p.148 of [START_REF] Viatcheslav | New Relations for the Euler Characteristic of Real Algebraic Manifolds[END_REF], the signature σ(CX) is congruent to (-1) k χ(CX) modulo 4. Therefore, the total Betti number of RX ε is congruent to χ(CX) modulo 4. Since CX only has homology in even dimensions χ(CX) equals its total Betti number.

Theorem 4.30 (Vanishing Criterion). Let r ≥ 2 be an integer congruent to n modulo 2. The differentials of the relevant page E r (RX ε ) are all zero if and only if i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective when q equals n-r 2 . Proof. Let r ≥ 2 and r = n (mod 2). By Theorem 4.26 the only possibly non-trivial differentials of the r-th page are:

d n+r 2 , n-r 2 -1 r and d n+r 2 -1, n+r 2 -1 r .
Following Theorem 4.27, they have the same rank so one vanishes if and only the other does. Let q be n-r 2 , we have the commutative diagram:

E q+r,q-1 r (RP ) E q+r,q-1 r (RX ε ) E q,q r (RP ) E q,q r (RX ε ) d q+r,q-1 r i q+r,q-1 r d q+r,q-1 r i q,q r (8)
Using Theorem 4.25, Theorem 4.26, and Proposition 4.24 the commutative diagram (8) can be written as follows: 0 E q+r,q-1 2 (RX ε ) H q,q (P ; F 2 ) H q,q (X; F 2 )

d q+r,q-1 r i q,q
Since 2q = n -r < n -1 by assumption, the tropical Lefschetz Hyperplane Section Theorem38 implies that i q,q is an isomorphism. As a consequence d q+r,q-1 r vanishes if and only if the map i q,q r+1 : E q,q r+1 (RP ) → E q,q r+1 (RX ε ) is injective. Furthermore, we have the following exact sequence:

0 → E q,q ∞ (RX ε ) → H q (RX ε ; F 2 ) → E n-1-q,q ∞ (RX ε ) → 0 .
Since E q,q ∞ (RX ε ) is isomorphic to E q,q r+1 (RX ε ), H q (RP ; F 2 ) is isomorphic to E q,q r+1 (RP ), and the morphism i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) respects the filtration, the morphism i q,q r+1 is injective if and only if i q is. From this vanishing criterion we deduce the following corollary: Corollary 4.31 (Criterion of Degeneracy). Let r 0 ≥ 2 be an integer congruent to n modulo 2. The differentials d p,q r of the Renaudineau-Shaw spectral sequence computing the cohomology of RX ε vanish for all p, q ≥ 0, and all r ≥ r 0 , if and only if the cohomological inclusions i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) are injective for all q ≤ n-r 0 2 . In particular, all the differentials d p,q r are trivial for all r ≥ 2 if and only if the cohomological inclusions i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) are injective for all q ≤ ⌊ n 2 ⌋ -1.

Corollary 4.31 can be interpreted as a comparison of two invariants of the pair RX ε ⊂ RP . The first one is the following: Definition 4.32. We define the degeneracy index of RX ε as: r(RX ε ) := min r 0 ≥ 0 | d p,q r = 0, ∀p, q ∈ N, ∀r ≥ r 0 .

The second invariant was introduced by I.O. Kalinin for projective hypersurfaces and named the rank of the hypersurface by O. Viro:

Definition 4.33. The rank of RX ε is defined as: ℓ(RX ε ) := max q 0 ≥ 0 | i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective for all q ≤ q 0 . Corollary 4.34. We have the inequalities:

ℓ(RX ε ) ≥ n -r(RX ε ) 2 , with equality if r(RX ε ) ≥ 3 + 1-(-1) n 2 and: r(RX ε ) ≤ max (2; n -2ℓ(RX ε ) -1) , with equality if ℓ(RX ε ) ≤ n-5 2 .
Proof. In a first time we prove the first inequality. The number r(RX ε ) lies between 1 and n -1. Assume r(RX ε ) ≥ 2 and let us denote by R and L the sets: R := 2 ≤ r 0 ≤ n | d p,q r = 0, ∀p, q ∈ N, ∀r ≥ r 0 , L := q 0 ≥ 0 | i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective for all q ≤ q 0 .

If we denote by

f the map r → n-r 2 , Corollary 4.31 implies that f (R) is a subset of L. Since f is non-increasing ℓ(RX ε ) is at least equal to f (min R) = f (r(RX ε )). If r(RX ε ) = 1
then the hypersurface RX ε is maximal for the Smith-Thom inequality 39 . There are two cases depending on the parity of n -1. If n -1 is odd then the inequality is a direct consequence of Corollary 4.31. If n -1 is even then we have to show that i n-1 2 is also injective. In the first page of the Renaudineau-Shaw spectral sequences we have the following commutative diagram:

E n+1 2 , n-3 2 1 (RP ) E n+1 2 , n-3 2 1 (RX ε ) E n-1 2 , n-1 2 1 (RP ) E n-1 2 , n-1 2 1 (RX ε ) i n+1 2 , n-3 2 1 d n+1 2 , n-3 2 1 d n+1 2 , n-3 2 1 i n-1 2 , n-1 2 1
39 c.f. [RS23] Theorem 6.1 p.26.

Using the maximality hypothesis as well as Theorem 4.25, Theorem 4.26, and Proposition 4.24 this commutative diagram can be written as follows:

0 H n+1 2 , n-3 2 (X; F 2 ) H n-1 2 , n-1 2 (P ; F 2 ) H n-1 2 , n-1 2 (X; F 2 ) 0 i n-1 2 , n-1 2 Where i n-1 2 , n-1
2 is injective by the tropical Lefschetz Hyperplane Section Theorem 40 . Since RX ε is maximal

H n-1 2 (RX ε ; F 2 ) is isomorphic to H n-1 2 , n-1
2 (X; F 2 ) and the morphism:

i n-1 2 : H n-1 2 (RP ; F 2 ) → H n-1 2 (RX ε ; F 2 ),
is conjugated to the morphism:

i n-1 2 , n-1 2 : H n-1 2 , n-1 2 (P ; F 2 ) → H n-1 2 , n-1 2 (X; F 2 ). a consequence i n-1 2 is also injective. If r(RX ε ) ≥ 3 + 1-(-1) n 2
then the differentials of at least one relevant page are non-trivial. Then, using Corollary 4.31 we find that the map i q is not injective for the corresponding index q and the equality follows. To prove the second inequality we use the converse implication of Corollary 4.31 to find that:

r(RX ε ) ≤ min r ≥ 2 | i q is injective for all q ≤ n-r 2 ≤ min r ≥ 2 | n-r 2 ≤ ℓ(RX ε ) ≤ min r ≥ 2 | n-r 2 ≤ ℓ(RX ε ) ∪ r ≥ 2 | n-r 2 ≤ ℓ(RX ε ) < n-r 2 ≤ min {max(2; n -2ℓ(RX ε ))} ∪ r ≥ 2 | n-r 2 ≤ ℓ(RX ε ) < n-r 2 . The set r ≥ 2 | n-r 2 ≤ ℓ(RX ε ) < n-r 2
is either empty or reduced to {n -2ℓ(RX ε ) -1}. The latter case only occurs when n -2ℓ(RX ε ) -1 ≥ 2 i.e. ℓ(RX ε ) ≤ n-3 2 . In that case we find r(RX ε ) ≤ n -2ℓ(RX ε ) -1. If on the contrary ℓ(RX ε ) ≥ n-1 2 then max(2; n -2ℓ(RX ε )) = 2 and r(RX ε ) ≤ 2. Therefore, r(RX ε ) is at most equal to max(2; n -2ℓ(RX ε ) -1). The associated equality follows from Corollary 4.31 since under the assumption ℓ(RX ε ) ≤ n-5

2 there is at least one relevant page of index at least 2 with non-trivial differentials.

Corollary 4.31 can be improved for a particular class of real toric varieties whose cohomology ring have a property related to the Hard Lefschetz Theorem 41 : Definition 4.35. Let Y be a topological space and α ∈ H 1 (Y ; F 2 ). We denote by ι(α) the minimum of the integers q ≥ -1 for which there exists a non-zero class β ∈ H q+1 (Y ; F 2 ) for which α ∪ β vanishes. We denote the maximum of the ι(α)'s for all α ∈ H 1 (Y ; F 2 ) by ι(Y ). i q : H q (RP ; F 2 ) → H q (RX ε ; F 2 ) is injective for q = n-r 0 2 . In particular, all the differentials d r p,q are trivial for all r ≥ 2 if and only if the cohomological inclusion i q : H q (RP ;

F 2 ) → H q (RX ε ; F 2 ) is injective for q = n 2 -1.
Proof. Let ω be a class of P for which ι(ω) ≥ n 2 -2. For all q ∈ N, we have the following commutative diagram:

H q (RP ; F 2 ) H q (RX ε ; F 2 ) H q+1 (RP ; F 2 ) H q+1 (RX ε ; F 2 ) i q ω∪- i 1 (ω)∪- i q+1
Therefore, if q + 1 ≤ ⌊ n 2 ⌋ -1 and i q+1 is injective then so is i q since ω ∪ -is injective. Corollary 4.38 follows from Corollary 4.31.

If P is a non-singular simplex of odd size and K is a convex triangulation, RX ε is isotopic to a real algebraic hypersurface of odd degree in a real projective space by Viro's Patchworking Theorem43 . In this case, it is known that ℓ(RX ε ) = n -1, hence all the differentials of E p,q r (RX ε ) p,q≥0 vanish for all r ≥ 2. This sufficient criterion of degeneracy can be generalised to the class of polytopes satisfying ι(RP ) ≥ n 2 -1 regardless of the convexity of the triangulation.

Proposition 4.39. We have the inequality:

ℓ(RX ε ) ≥ ι[ω RX ] .
In particular it follows from Corollary 4.38 that if ι[ω RX ] ≥ n 2 -1 then all the differentials of E p,q r (RX ε ) p,q≥0 vanish for all r ≥ 2.

Proof. For all 0 ≤ q ≤ n -1, we have the following commutative diagram:

H q (RP ; F 2 ) H n-q (RP ; F 2 ) H n-1-q (RP ; F 2 ) H n-1-q (RX ε ; F 2 ) H q (RX ε ; F 2 ) -∩[RP ] [ω RX ]∩- i n-1-q -∩[RXε] i q
Where ∩ denotes the cap product,

H q (RP ; F 2 ) H q+1 (RP ; F 2 ) H n-q (RP ; F 2 ) H n-q-1 (RP ; F 2 ) [ω RX ]∪- -∩[RP ] ∼ = -∩[RP ] ∼ = [ω RX ]∩- Therefore, the morphism i q is injective for all q ≤ ι[ω RX ], and ℓ(RX ε ) ≥ ι[ω RX ].
Corollary 4.40. The Renaudineau-Shaw spectral sequence of a T-hypersurface of odd degree in a projective space degenerates at the second page.

Combining Corollary 4.34 and Proposition 4.39 yields a generalisation in some cases of the statement "The Renaudineau-Shaw spectral sequence of a projective primitively patchworked hypersurface of odd degree degenerates at the second page", namely the upper bound:

r(RX ε ) ≤ max (2; n -2ι[ω RX ] -1) .
Therefore we can sometimes directly know if the Renaudineau-Shaw spectral sequence degenerates at the second page by only looking at the degree of RX ε , i.e. at the class [ω RX ]. In Table 1 and Table 2 below we described the the possible values of ι[ω RX ] and the associated upper bound on the degeneracy index. In the first table we study the case of a product of non-singular simplices and in the second one the case of a blow-up of a non-singular simplex.

Both these examples have a H 1 of dimension 2. However for some polytopes, e.g. a cube of dimension at least 4, this upper bound is completely vacuous no matter the degree of the hypersurface.

Degeneracy for a Family of Triangulations

In this section we will construct a particular sequence of primitive triangulations of the nonsingular simplices inspired from the triangulations on which I. Itenberg constructs44 maximal surfaces of every degree. We will then show that any T-hypersurface constructed on these

P n d × P n+m d ′ d ′ even d ′ odd d even 0 n + m -1 d odd n -1 n + m -1 (
] • E = 0 [RX ε ] • E = E 2 d even 0 0 d odd 0 n -2 (a)
The value of ι[ω RX ] in the blow-up of an n-projective space as a function of the parity of the degree d and the intersection with the exceptional divisor E.

Bl.P n d [RX ε ] • E = 0 [RX ε ] • E = E 2 d even max(2; n -1) max(2; n -1) d odd max(2; n -1) 2 (b)
The corresponding upper bound on r(RX ε ) deduced from Corollary 4.34 and Proposition 4.39.

Table 2: The value of ι[ω RX ] as well as the corresponding upper bound on the degeneracy index of a T-hypersurface in the blow-up as a function of the degree and the intersection with the exceptional divisor.

triangulations has degeneracy index equal to 2. To do so we will prove a stronger statement than the criterion of Corollary 4.38. We begin with two auxiliary propositions.

Proposition 5.1. Let K be a finite simplicial complex, ΣK its suspension and ε ∈ C 0 (ΣK; F 2 ). We denote by a + and a -the two suspension apexes. If ε(a + ) + ε(a -) equals 1 then the hypersurface X dual to dε is isotopic to |K| relatively to |K| ∩ X.

Proof. The hypersurface X is the support of the subcomplex of the cubical subdivision of ΣK made of the cubes indexed by the pairs of simplices σ 1 ≤ σ n for which dε(σ 1 ) = 1. The geometric realisation of ΣK lives in the finite dimensional real vector space freely generated by the vertices of ΣK and the hypersurface X is homeomorphic to the intersection of |ΣK| with the hyperplane of equation:

f := p∈ΣK (0) (-1) ε(p) x p = 0 ,
where x p is the coordinate associated with the vertex p. We can construct a homeomorphism G of |ΣK| that exchanges the two sets the following way:

1. Let σ ∈ ΣK, if the restriction ε| σ is non-constant we can find a set of positive real numbers u(σ) indexed by the vertices of σ whose sum is 1 and satisfying:

p∈σ (0) (-1) ε(p) u p (σ) = 0 .
If the restriction ε| σ is constant we set u p (σ) = 1 dim σ+1 ; 2. We denote by G the homeomorphism of |ΣK|, affine on the simplices of the barycentric subdivision of ΣK, which sends the barycenter of every simplex σ to p∈σ (0) u p (σ)p.

This map G is simplicial by construction. Now we study the hypersurface of |ΣK| defined by f . It avoids the two apexes a + and a -so |ΣK| ∩ {f = 0} equals |ΣK| \ {a ± } ∩ {f = 0}. Let us consider the following homeomorphism:

φ : |K|×] -1; 1[ -→ |ΣK| \ {a + ; a -} (p; t) -→ (1 -|t|)p + max(0; t)a + -min(0; t)a -. Since ε(a + )+ε(a -) equals 1, f (φ(p; t)) is equal to (1-|t|)f (p)+(-1) ε(a + ) t. If f (φ(p; t)
) vanishes and f (p) does not, 0 < |t| < 1. And therefore:

t |t| = (-1) ε(a + )+1 f (p) |f (p)| .
From which we deduce, after considering the case f (p) = 0, that f (φ(p; t)) vanishes if and only if:

f (p) + (-1) ε(a + ) (1 + |f (p)|)t = 0 .
The hypersurface {f = 0} is the graph of the function u : p → (-1) ε(a + )+1 f (p) 1+|f (p)| in the coordinates given by φ. We consider the map H : [0; 1]×|K| → |ΣK| defined for all (t; p) ∈ [0; 1]×|K| by the formula: H(t; p) := G -1 (1 -t)G(p) + tφ G(p); u(G(p)) .

The map H(t; -) is an embedding for all 0 ≤ t ≤ 1 since it is conjugated to a graph embedding, H(0; -) is the inclusion of |K| in |ΣK|. The set H(1; |K|) is X. If p is in |K| ∩ X then G(p) belongs to |K| ∩ {f = 0}. Consequently, H(t; p) equals G -1 (1 -t)G(p) + tG(p) which is p.

We denote by B n+1 the (n + 1)-dimensional Euclidean ball and by π the projection of B n+1 onto RP n+1 by antipodal identification of the points of ∂B n+1 . Let X be a subset of RP n , we define CX ⊂ RP n+1 as the image under π of the cone at the origin 0 over π -1 (X).

Lemma 5.2. If X ⊂ RP n is homeomorphic to RP p and homologous to a linear sub-space then CX is homeomorphic to RP p+1 and homologous to a linear sub-space as well.

Proof. The space π -1 (X) is a double cover of X. If it is connected and p > 1 π -1 (X) → Xis the universal cover of X and π -1 (X) is a sphere. If p = 1 and it is connected π -1 (X) is a circle. In this case CX is homeomorphic to RP p+1 . Now we prove that if X is homologous to a linear sub-space of RP n then π -1 (X) is connected. We have the commutative diagram of exact sequences of singular chain complexes: where the vertical morphism are given by the corresponding inclusions of topological spaces and the morphism i sends a singular simplex σ to the sum of its two lifts to S n . It yields the following commutative diagram of exact sequences:

0 H 0 (RP n ; F 2 ) H 0 (S n ; F 2 ) H 0 (RP n ; F 2 ) H 1 (RP n ; F 2 ) • • • 0 H 0 (X; F 2 ) H 0 (π -1 (X); F 2 ) H 0 (X; F 2 ) H 1 (X; F 2 ) • • • ∼ = 0 ∼ = ∼ = π * i ∼ = j *
It implies in particular that b 0 (π -1 (X); F 2 ) = 1 + rk i = 2 -rk j * . By assumption the image of the fundamental class of X in the homology of RP n is [RP p ] so its intersection with [RP n+1-p ] is the class of the line [RP 1 ]. If α is the Poincaré dual of [RP n+1-p ], we have:

j * (j * α ∩ [X]) = [RP n+1-p ] • [RP p ] = [RP 1 ],
so j * has rank 1 and π -1 (X) is connected.

To see that CX is still homologous to a linear sub-space we can notice that we have the following commutative square:

H 1 (RP n+1 ; F 2 ) H 1 (RP n ; F 2 ) H 1 (CX; F 2 ) H 1 (X; F 2 ) ∼ = ∼ = ∼ =
So the generator of H 1 (RP n+1 ; F 2 ) is sent to the generator of H 1 (CX; F 2 ). Using the structure of the cohomology rings of projective spaces, we see that the restriction morphism:

H q (RP n+1 ; F 2 ) → H q (CX; F 2 ), is an isomorphism for all q ≤ p + 1.

(v 0 ; 0) (v 2 ; 0) (v 1 ; 0) (v 0 ; 1) (v 2 ; 1) (v 1 ; 1) We recall that for a simplex σ n whose vertices are ordered we have a canonical triangulation of the prism σ n × [0; 1]. Its maximal simplices are the joins [v 0 ; ...; v i ] × {1} * [v i ; ...; v n ] × {0} where 0 ≤ i ≤ n and [v 1 ; ...; v k ] is the face of σ indexed by the vertices v 1 , ..., v k . See Figure 7 for an example. This triangulation allows us to construct triangulations of the standards simplices inductively.

Definition 5.3. Let n, d ≥ 1 be two integers and P n d be the convex hull of {0; de 1 ; ...; de n } ⊂ R n . If K is a triangulation of P n d and L is a triangulation of P n-1 d+1 we define K + L as the unique triangulation of P n d+1 for which:

1. The restriction of K + L to P n d+1 ∩ {x n ≥ 1} is the translation of K; Lemma 5.4. Let P be a full dimensional polytope of R n endowed with a triangulation K, and f, g : P → R be two functions, affine on every simplex of K. If f is convex and, for any convex subset A of P , g| A is convex as soon as f | A is affine, then there is an η > 0 for which f + ηg is convex.

Proof. Let σ be a maximal simplex of K, for all functions h : P → R affine on every simplex of K, we denote by a(h; σ) the unique affine function of R n satisfying h| σ = a(h; σ)| σ . For

The intersection of RX ε with this RP 2n+1 d is, by construction, the patchwork of the restriction of ε to this codimension 2 face of P 2n+3 d . This face is the codimension 1 of the bottom face opposite to the origin. Since the bottom face is endowed with the Viro triangulation, Proposition 5.8 implies that the triangulation of the codimension 2 face is the Viro triangulation V 2n+1 d . Hence we can use our recursion hypothesis to find a sub-space L n ⊂ RX ε ∩ RP 2n+1 d homeomorphic and homologous to RP n . Following Lemma 5.2 we know that CL n ⊂ RP 2n+3 d is homeomorphic and homologous to RP n+1 . The preimage π -1 (CL n ) is located in the equatorial ball B 2n+2 of B 2n+3 , the one spanned by the codimension 1 face of P 2n+3 d opposite to the vertex de 2n+3 . From Proposition 5.1 we can move the ball B 2n+2 inside π -1 (RX ε ) by isotopy relative to B 2n+2 ∩ π -1 (RX ε ). Indeed, the union of the (2n + 3)-dimensional simplices of RV 2n+3 d that have a codimension 1 face in B 2n+2 is the suspension of the triangulation of B 2n+2 . Moreover, the two suspension apexes are ±e 2n+3 which carry opposite signs by definition. The isotopy sends π -1 (CL n ) to a sub-space that projects to L n+1 ⊂ RX ε which satisfies the desired properties. The isotopy unfolding of the portion of RX ε located inside the suspension of the equatorial ball is depicted in Figure 12. From this we deduce that if RX ε has even dimension 2n we have, for all q ≥ 0, a commutative diagram:

H q (RX ε ; F 2 ) H q (RP 2n+1 ; F 2 ) H q (L n ; F 2 ) iq jq
The map j q is surjective for all 0 ≤ q ≤ n, so the theorem is proven for even dimensional patchworks. For odd dimensional patchworks we can notice that the triangulation restricted to the codimension 1 face opposite to de 2n+2 carries the Viro triangulation as well so we can apply our construction to the induced patchwork and deduce the statement.

Combining this statement with Corollary 4.38 we find the following corollary:

Corollary 5.10. The Renaudineau-Shaw spectral sequences computing the homology and the cohomology of the hypersurface RX ε ⊂ RP n constructed from a Viro triangulation K and a sign distribution ε ∈ C 0 (K; F 2 ) degenerates at the second page.

A. Renaudineau and K. Shaw Conjecture can be rephrased as r(RX ε ) ≤ 2 or equivalently ℓ(RX ε ) ≥ n 2 -1. However, as in Theorem 5.9, we believe the stronger inequality ℓ(RX ε ) ≥ n-1 2 might even be true in full generality. (f) The portion of RX ε inside the suspension at t = 1.

(g) The portion of RX ε inside the suspension at t = 0 from above.

(h) The portion of RX ε inside the suspension at t = 1 2 from above.

(i) The portion of RX ε inside the suspension at t = 1 from above.

  (a) The dual hypersurface of a triangulation of a triangle. (b) Two views of the dual hypersurface of a tetrahedron.

Figure 2 :

 2 Figure 2: Two examples of dual hypersurfaces.

defined by the formula: ∂c = e k+1 ∈E e k ≤e k+1 c e k+1 e k+1 e k ⊗

 k e k , where -| e k+1 e k

  and all cells e k+1 , by the formula: dα(e k+1 ) = e k ≤e k+1 α(e k ) e k+1 e k . where -| e k+1 e k

  The triangulation K and the sign distribution ε. the associated T-curve RX ε .

Figure 4 :

 4 Figure 4: An example of T-curve in P 1 (R) × P 1 (R).

  Lemma 4.2 ([BFMvH06] Lemma 6.2 p.8, and [RS23], Lemma 4.1 p.21). For all k ∈ N, the ideal m k V is spanned, as sub-vector space, by the following elements: w∈W x w , 30 c.f. [BFMvH06] Lemma 6.2 p.8 or [RS23] Lemma 4.1 p.21

Definition 4. 8 .

 8 The functor O

  Definition 4.20 (A. Renaudineau and K. Shaw, [RS23] Definition 4.5 p.

Figure 5 :

 5 Figure 5: Some page of the spectral sequence of a T-hypersurface of dimension 3.

  since n -1 -n+r 2 + r equals n+r 2 -1, and n -2 -n-r 2 + 1 equals n+r 2 -1.It follows that V is divisible by 4. Using the same argument we get that the differential d p,q

  40 c.f. [ARS21] Theorem 1.1 p.1348 and [BM71] Proposition 3.2 p.15. 41 c.f. [Wel07] Corollary 4.13 p.195.

Figure 6 :

 6 Figure 6: A blow-up of P 1 × P 1 .

0

  C * (RP n ; F 2 ) C * (S n ; F 2 ) C * (RP n ; F 2 ) 0 0 C * (X; F 2 ) C * (π -1 (X); F 2 ) C * (X; F 2 ) 0

Figure 7 :

 7 Figure 7: Triangulation of the Product of a Segment with a Triangle.

2.

  The restriction of K + L to [e n , e n + de 1 , ..., e n + de i ] * [(d + 1)e i , ..., (d + 1)e n-1 ], for all 0 ≤ i ≤ n-1, is the join of the restriction of the translation of K to [e n , e n +de 1 , ..., e n +de i ] with the restriction of L to [(d + 1)e i , ..., (d + 1)e n-1 ].

Figure 8 Figure 8 :

 88 Figure 8 depicts an example of such triangulation.

  (a) The suspension of the equatorial ball at t = 0. (b) The suspension of the equatorial ball at t = 1 2 . (c) The suspension of the equatorial ball at t = 1. (d) The portion of RX ε inside the suspension at t = 0.(e) The portion of RX ε inside the suspension at t = 1 2 .

Figure 12 :

 12 Figure 12: The unfolding isotopy from Proposition 5.1 applied to the portion of a T-surface RX ε on V 3 4 located in the suspension of the equatorial ball.

  [RP ] is the generator of H n (RP ; F 2 ), and [RX ε ] is the generator of H n-1 (RX ε ; F 2 ). Because both RP and RX ε satisfy the Poincaré duality with F 2 -coefficients, both -∩ [RP ] and -∩ [RX ε ] are isomorphisms. We also have the following commutative diagram:

  a) The value of ι[ω RX ] in the product of an n-projective space with an (n + m)projective space as a function of the parity of the bidegree (d; d ′ ).

	P n d × P n+m d ′	d ′ even	d ′ odd
	d even	max(2; n + m -1)	2
	d odd	max(2; m + 1)	2
	(b) The corresponding upper bound on r(RX ε ) de-
	duced from Corollary 4.34 and Proposition 4.39.

Table 1 :

 1 The value of ι[ω RX ] as well as the corresponding upper bound on the degeneracy index of a T-hypersurface in a product of two projective spaces as functions of the bidegree.

	Bl.P n d	[RX ε

c.f. [JRS18] Theorem 5.3 p.16.

c.f. [ARS21] Theorem 1.1 p.1348 and [BM71] Proposition 3.2 p.15.

c.f.[START_REF] Mccleary | A User's Guide to Spectral Sequences[END_REF] Theorem 2.6 p.33.

c.f. [CE56] Theorem 3.3a p.114.

c.f.[START_REF] Deligne | Théorie de Hodge[END_REF] (1.1.7) p.7.

c.f. [CE56] Theorem 3.3a p.114.

c.f.[START_REF] Fulton | Introduction to Toric Varieties[END_REF] Proposition p.29.

c.f. [IKMZ19] Definition 13 p.10.

i.e. the interior of σ p in the affine space it spans.

we use the standard notation e I := e i1 ∧ ... ∧ e i k where I = {i 1 < ... < i k }.

c.f. [CE56] Theorem 3.3a p.114.

c.f.[START_REF] Daniel G Quillen | On the Associated Graded Ring of a Group Ring[END_REF].

c.f. Definition 1.3 or[START_REF] Deligne | Théorie de Hodge[END_REF] (1.1.6) p.7.

a 2 = a for all a ∈ F 2

c.f. [Lan93] Lemma 9.1 p.158.

c.f. Proposition 4.12.

c.f. [CE56] Theorem 3.3a p.114.

c.f.[START_REF] Bertrand | Euler Characteristic of Primitive T-Hypersurfaces and Maximal Surfaces[END_REF] Theorem 2.1 p.12 and [BLdMR22] Theorem 1.3 p.4.

c.f. [ARS21] Theorem 1.1 p.1348 and [BM71] Proposition 3.2 p.15.

c.f.[START_REF] Munkres | Elements of Algebraic Topology[END_REF] Theorem 33.1 pp.186-187.

c.f. [Vir06] Theorem 4.3.A pp.32-33, or [Ris93] Théorème 4.2 p.75.

c.f.[START_REF] Itenberg | Topology of Real Algebraic T-surfaces[END_REF], §5 M-Surfaces pp.139-141.

This research was funded, in whole or in part, by l'Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0014. For the purpose of open access, the author has applied a CC-BY public copyright licence to any Author Accepted

Remark 4.36. For any n-dimensional smooth polytope P the number ι(RP ) is at most n -1. The Hard Lefschetz Theorem might lead us to think that ι(RP ) is always at least equal to n 2 -1 since the cohomology ring of RP with F 2 -coefficients is isomorphic, up to dividing the grading by 2, to the reduction modulo 2 of the integral cohomology ring of its complex locus CP . However, the powers of the class of a primitive ample line bundle might be divisible by 2 from a certain rank. This phenomenon is illustrated in the fourth point of Examples 4.37. Nevertheless, we can deduce from the Hard Lefschetz Theorem that ι(RP ) being greater than n 2 -1 is a constraint on the polytope P . Since the Betti numbers of RP form an unimodal sequence centered at n 2 , if ι(RP ) ≥ n 2 then all the q-th Betti numbers of RP must be equal for all n -ι(RP ) -1 ≤ q ≤ ι(RP ) + 1, as in the third point of Examples 4.37. The only polytope P satisfying ι(RP ) = n -1 is the simplex.

Examples 4.37. Here we give examples of the numbers ι(Y ) when Y is the real locus of a smooth projective toric variety:

1. If P is a non-singular simplex its associated toric variety is a projective space. The cohomology ring of RP is isomorphic to

where n is the dimension of the simplex. We have ι(RP ) = n -1;

2. If P is the product of two non-singular simplices then the cohomology ring associated with RP is

3. We say that a simple integer polytope P is a blow-up of a simple integer polytope Q if P is obtained by chopping off one of its corner. More precisely, the corner to be chopped off must be adjacent to edges of integral length at least 2 and the chopping hyperplane must pass through integer points of these edges at equal integer distances from the corner. See Figure 6 for instance. The corner vertex of the polytope corresponds to a fixed point of the torus action on the associated toric variety. Chopping off the corner corresponds to blowing-up the fixed point. One can find a description of the normal fan on pp.40-41 of [START_REF] Fulton | Introduction to Toric Varieties[END_REF]. The space RP is the connected sum of RQ with RP n . Using the Mayer-Vietoris exact sequence 42 we find that q∈N H q (RP ;

However, the class α, seen as a class of P , has ι-number equal to either 0 or -1. Since x has ι-number equal to 0 we find that ι(RP ) = min(ι(RQ); n -2).

4. Let I n be the n-dimensional cube. The cohomology ring q∈N H q (RI n ; F 2 ) is the exterior algebra H 1 (RI n ; F 2 ) so any class of degree 1 squares to 0. Therefore, ι(RI n ) = 0.

Corollary 4.38. Let r 0 ≥ 2 be an integer congruent to n modulo 2. If ι(RP ) is at least n 2 -2 then the differentials d r p,q of the Renaudineau-Shaw spectral sequence computing the cohomology of RX ε vanish for all p, q ∈ N, and all r ≥ r 0 , if and only if the cohomological inclusion all vertices v of K and all maximal simplices σ of K, we denote by δ(h; v; σ) the difference h(v) -a(h; σ)(v). The function h is convex if and only if:

which is itself equivalent to δ(h; v; σ) ≥ 0 for all vertices v and maximal simplices σ. Moreover, if h is convex then δ(h; v; σ) vanish if and only if v and σ belong to the same affinity component of h. Indeed, since h is convex, its affinity components are the projections of the n-dimensional faces of the epigraph of h, i.e. the sets {p ∈ P | h(p) = a(h; σ)(p)} where σ is a maximal simplex of K. By assumption, we have δ(g; v; σ) ≥ 0 whenever δ(f ; v; σ) = 0. If min{δ(g; v; σ) : v, σ ∈ K} ≥ 0 the function g is convex and f + ηg is convex for all η > 0. In the other case f cannot be affine and there is at least one couple (v; σ) for which δ(f ; v; σ) is positive, and any η satisfying:

yields a convex function f + ηg, since δ(f + ηg; v; σ) = δ(f ; v; σ) + ηδ(g; v; σ) ≥ 0 for all vertices v and all maximal simplices σ.

Proposition 5.5. If both K and L are convex then so is K + L.

Proof. The triangulation K+L of P n d+1 is a subdivision of the first triangulation M of P n d+1 made of the translation of P n d by e n and the joins of the second point of Definition 5.3 subdividing the prism P n d+1 ∩ {0 ≤ x n ≤ 1}. The triangulation M is convex. It has exactly (n + 1) maximal simplices, the interior of each is a connected component of the complement in P n d+1 of a union of n affine hyperplanes. A direct computation yields their equations: (d + 1)(x n -1) + n-1 k=i x k , for all 1 ≤ i ≤ n -1, and x n = 1. On that account, the function:

certifies the convexity of M . Let ν K and ν L be two functions respectively certifying the convexity of K and L. We consider [ν K + ν L ] the unique function P n d+1 → R affine on every simplex of K + L whose restriction to P n d+1 ∩ {x n ≥ 1} is the translation of ν K by e n and whose restriction to P n d+1 ∩ {x n = 0} is ν L . In other words [ν K + ν L ] is defined on the joins of the bottom prism P n d+1 ∩ {0 ≤ x n ≤ 1} by the convex interpolation of ν K (-+ e n ) and ν L . By construction this function is convex on the simplices of M , therefore, following Lemma 5.4 we can find η > 0 small enough so that ν + η[ν K + ν L ] is convex. By construction, this function certifies the convexity of K + L. Proposition 5.8. Let n ≥ 2 and d ≥ 1. The injective affine map:

that sends 0 to de 1 and de i to de i+1 , for all

Proof. The proposition is a consequence of the recursive construction of the Viro triangulations.

Let us denote

1 is the trivial triangulation, and K 1 d is the unique primitive triangulation of the segment. Therefore K n d is V n d for all d, n ≥ 1.

Theorem 5.9 (Rank Maximality). If K is a Viro triangulation of P n then the homological inclusion:

is surjective for all q ≤ [ n-1 2 ] and all ε ∈ C 0 (K; F 2 ). That is to say, the rank of a projective hypersurface obtained from a primitive patchwork on a Viro triangulation is always maximal.