Rémi Bignalet
email: bignalet@dima.unige.fr

Rémi Bignalet-Cazalet

Gluing

Determinantal Gluing

Émi Cremona

Bignalet-Cazalet

Gluing determinantal maps

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In this note, we consider rational maps f = (f 0 : . . . : f n ) : P n k P n k defined by n + 1 homogeneous polynomials f 0 , . . . , f n ∈ R = k[x 0 , . . . , x n ] without common factor, that are determinantal i.e. that the base ideal I f = (f 0 , . . . , f n ) ⊂ R of f is the n-minors ideal of its minimal presenting matrix Φ f .

Our guiding remark is that the so-called standard Cremona map (see [START_REF] Gonzalez-Sprinberg | On characteristic classes of determinantal Cremona transformations[END_REF] for more references about the standard Cremona maps)

τ n = (x 1 • • • x n : x 0 x 2 • • • x n : . . . : x 0 • • • x n-1 ) : P n k P n
k can be interpreted as the gluing of n standard Cremona maps τ

(1) 1 = (x 1 : x 0 ) :

P 1 k P 1 k , τ (2) 1
= (x 2 : x 1 ) :

P 1 k P 1 k , . . ., τ (n) 1
= (x n : x n-1 ) : P 1 k P 1 k . Indeed, I τn is the n-minors ideal of the (n + 1) × n matrix completed with zeros entries. This work takes it roots in trying to generalize this previous construction and, given two integers m, m ′ 1, to understand the projective degrees of such an outputted glued determinantal map [g|g ′ ] : P m+m ′ k P m+m ′ k via the properties Date: May 9, 2021. 2010 Mathematics Subject Classification. 13D02, 14E05, Key words and phrases. determinantal Cremona maps, multidegree/projective degrees of a rational map, mixed volumes of polytopes, Bernstein's theorem on sparse polynomial systems, glued determinantal Cremona map.

Φ τn =           x0 0
The author was funded by the European Union's Horizon 2020 research and innovation program as a Marie Sk lodowska-Curie fellow of the Istituto Nazionale di Alta Matematica "Francesco Severi" grant No. 713485.

of the inputted determinantal maps g : P m k P m k and g ′ : P m ′ k P m ′ k . Recall that given any map f = (f 0 : . . . : f n ) : P n k P n k and assuming that the base field k is algebraically closed, the projective degrees d 0 (f ), d 1 (f ) . . . , d n (f ) of f are the quantities such that for any k ∈ {0, . . . , n}, d k (f ) is the cardinal # H k

x ∩ f -1 (H n-k y ) of the intersection between a general linear space H k

x of codimension k in the source space of f with the preimage of a general linear space H n-k y of codimension nk in the target space of f , see [START_REF] Harris | Algebraic Geometry[END_REF]Ex. 19.4] or below for more developments about the projective degrees of a map. Given this latter definition, let us state a first result describing, in some situations, the projective degree of [g|g ′ ].

Proposition 1. Let m, m ′ 1, R m = k[x 0 , . . . , x m ], R m ′ = k[x m , .
. . , x m+m ′ ] and R = k[x 0 , . . . , x m+m ′ ] and let g : P m k P m k (resp. g ′ : P m ′ k P m ′ k ) be a determinantal map such that I g (resp. I g ′ ) is the m-minors ideal of a matrix

Φ g ∈ R (m+1)×m m (resp. the m ′ -minors ideal of Φ g ′ ∈ R (m ′ +1)×m ′ m ′ ). Put                 Φ g 0 m×m ′ 0 m ′ ×m Φ g ′ m m ′ + 1 m m ′ m + 1 m ′ Φ [g|g ′ ] = ∈ R (m+m ′ +1)×(m+m ′ )
and assume that the ideal of (m + m ′ )-minors of Φ [g|g ′ ] is the base ideal of a map [g|g ′ ] : P m+m ′ k P m+m ′ k . Suppose that the tensor product F g ⊗ F g ′ of a free resolution F g of the graph of g and a free resolution F g ′ of the graph of g ′ provides a free resolution of the graph of [g|g ′ ]. Then, given any k ∈ {0, . . . , m + m ′ } (1)

d k ([g|g ′ ]) = k p=0 d p (g)d k-p (g ′ )
with the convention that d p (g) = 0 (resp.

d p (g ′ ) = 0) if p > m (resp. p > m ′ ).
As we will explain, Proposition 1 applies in particular in the previous example of the standard Cremona map.

Actually, other definitions of a glued determinantal map [g|g ′ ] can be given depending on the convention adopted to concatenate the matrices Φ g and Φ g ′ . For instance, in the case where each entry of any given column of Φ g and Φ g ′ is a general combination of given polynomials, a pseudo concatenate matrix Φ [g|g ′ ] and an associated glued map [g|g ′ ] can still be defined. In this direction:

Proposition 2.14. Let m, d 1 and let Φ [g|g ′ ] = (φ ij ) 0 i m+2 1 j m+2
be such that:

• for all l ∈ {1, . . . , m}, all the entries φ il of the l-th column of Φ [g|g ′ ] are general linear combinations of x 0 , . . . , x m , • all the entries φ i,m+1 of the m + 1-th column of Φ [g|g ′ ] are general linear combinations of x m+1 and x m+2 ,

• all the entries φ i,m+2 of the m + 2-th column of Φ [g|g ′ ] are general linear combinations of the generators of the ideal

(x m , x m+1 , x m+2 ) • (x m+1 , x m+2 ) d-1 = (x m x d-1 m+1 , x m x d-1 m+2 ) + (x m+1 , x m+2 ) d .
Then the map [g|g ′ ] :

P m+2 k P m+2 k whose base ideal I [g|g ′ ] is the (m+2)-minors ideal of Φ [g|g ′ ]
is a determinantal map and moreover:

∀k ∈ {0, . . . , m + 2}, d k ([g|g ′ ]) = m m -k + (d + 1) m m -k + 1 + m m -k + 2 with the convention that j i = 0 if i < 0 or i > j. In particular, the projective degrees' vector d([g|g ′ ]) = d 0 ([g|g ′ ]), . . . , d m+2 ([g|g ′ ]) of [g|g ′ ] is palindromic.
Our goal behind this glued construction is to approach combinatorially the projective degrees of the almost linear determinantal maps, i.e. the determinantal map f defined by almost linear matrix Φ f for which all but one column are filled with linear polynomials (see [START_REF] Kustin | Rational normal scrolls and the defining equations of Rees algebras[END_REF] for development about maps defined by almost linear matrices), in order to eventually extend in greater dimension previous works such as in [START_REF] Déserti | Quarto-quartic birational maps of P 3 (C)[END_REF].

The proof of Proposition 2.14 relies on the following observations. When considering a map f : P n k P n k of base ideal I f with presenting matrix Φ f , the scheme P(I f ) = Proj Sym(I f ) provides an approximation of the projective degrees of f . This is now a classical approach, see [START_REF] Russo | On birational maps and jacobian matrices[END_REF], [START_REF] Busé | Torsion of the symmetric algebra and implicitization[END_REF], [START_REF] Busé | Degree and birationality of multi-graded rational maps[END_REF] and [START_REF] Cid-Ruiz | Mixed multiplicities and projective degrees of rational maps[END_REF] for developments and applications in broader contexts. If f is in addition Koszul-determinantal i.e. P(I f ) is a complete intersection in its embedding in P n k × P n k defined by Φ f (a condition on f stronger than being determinantal and that we will explicit below), one can use the resolution of zero-dimensional complete intersection defined by sparse polynomials to compute, in some situations, the projective degrees of f . Indeed, Bernstein's theorem states that, under conditions, the mixed volume associated to a polynomial system (E) compute the solutions of (E) with non zero coordinates, see [CLO05, 7. Th. 5.4] or Theorem 2.7 below. Proposition 2.14 then follows from convex geometry, and in particular the projection formula in [ST10, Lemma 6] decomposing some mixed volumes as the product of mixed volumes in smaller dimension.

Contents of the paper. In Section 1, we precise the definition of the projective degrees of a map f , emphasizing how they are related to (bi-graded) free resolutions of the ideal of the graph Γ f of f . It gives the background of our proof of Proposition 1. In Section 2, we precise the approximations of the projective degrees of a Koszul-determinantal map given by the mixed volumes associated to the presentation matrix its the base ideal. We then give another definition of a glued determinantal map [g|g ′ ] :

P m+m ′ k P m+m ′ k via general properties of two initial determinantal maps g : P m k P m k and g ′ : P m ′ k P m ′ k .
Under assumptions on g and g ′ , the projective degrees of such a glued map [g|g ′ ] are then described by a Kunneth-like formula in the projective degrees of g and g ′ , see Proposition 2.14 and its proof.

The explicit computations given in this paper were made using basic functions of the software systems Polymake and Macaulay2 and the Cremona package [START_REF] Stagliano | A Macaulay2 package for computations with rational maps[END_REF] associated. The corresponding codes are available on request.

Acknowledgements. I thank Adrien Dubouloz and Daniele Faenzi for their comments about a first version of this manuscript. I also warmly thank Alessandro de Stefani for its help on some arguments of the first section.

Notation 2. We let k be any field in Section 1 and k = C in Section 2.

When recalling generalities about rational maps, we consider a positive integer n ∈ N * , R = k[x 0 , . . . , x 0 ], P n k = Proj(R) and a rational map f :

P n k P n k . When considering glued maps, we let m, m ′ 1, R m = k[x 0 , . . . , x m ], R m ′ = k[x m , . . . , x m+m ′ ], R = k[x 0 , . . . , x m+m ′ ]
and we always consider that R m and R m ′ are embedded in R. Let also g :

P m k P m k (resp. g ′ : P m ′ k P m ′ k )
, where

P m k = Proj(R m ) (resp. P m ′ k = Proj(R m ′ )).
The associated glued map is denoted [g|g ′ ] :

P m+m ′ k P m+m ′ k where P m+m ′ k = Proj(R m+m ′ ).

Free resolutions of the graph of glued determinantal maps

Our motivation in describing a free resolution of the ideal of the graph of a map f (free resolution of the graph for short) lies on the fact it gives an insight on the projective degrees of f and let us briefly outline why.

As explained in [MS05, Chapter 8], the theory of multigraded Hilbert series is the fine context for defining and studying the multidegree of subschemes in multiprojective spaces so we adapt here previous expositions as in [START_REF] Miller | Combinatorial Commutative Algebra[END_REF] or the more recent one in [START_REF] Cid-Ruiz | Mixed multiplicities and projective degrees of rational maps[END_REF] to the following bi-graded context. Let n 1 be an integer, k be any field (if no additional specific assumptions), R = k[x 0 , . . . , x n ] and S = R[y 0 , . . . , y n ] be the Z 2 -graded polynomial ring with the standard graduation deg(x i ) = (1, 0) and deg(y i ) = (0, 1) for all i ∈ {0, . . . , n}. Given a bi-graded algebra B, we let BiProj(B) be the set of bi-graded prime ideals of B. Thus, a subscheme X of BiProj(S) ≃ P n k × P n k is defined by a bi-graded ideal A ⊂ S and one associates to X = BiProj(S/A) its bi-variate Hilbert series

H S/A (T 0 , T 1 ) = (n0,n1)∈Z 2 (length(S/A) (n0,n1) T n0 0 T n1 1 .
Then, writing H S/A (T 0 , T 1 ) = Num S/A (T0,T1)

(1-T0) l (1-T1) l and assuming that X = BiProj(S/A) has codimension c, the coefficients of the homogeneous component of Num S/A (1 -T 0 , 1 -T 1 ) of total degree c define the multidegree of X.

Definition 1.1 (multidegree of a subscheme in P n k ×P n k ). Let A be a bi-graded ideal of S and assume that X = BiProj(S/A) has codimension c in BiProj(S)

≃ P n k × P n k . Given k ∈ {0, . . . , c}, define deg c-k,k P X as the coefficient of the monomial of bi-degree (c -k, k) of Num S/A (1 -T 0 , 1 -T 1 ).
The multidegree deg P X of X is the c + 1-uple deg P X = deg c,0 P X, . . . , deg 0,c P X . Remark 1.2. As explained for instance in [START_REF] Harris | Algebraic Geometry[END_REF]Proposition 7.16] or [CR21, Theorem 4.7], the multidegree deg P X = deg c,0 P X, . . . , deg 0,c P X of a c-dimensional subscheme X ⊂ P n k × P n k has the following geometric interpretation. Assuming the base field k is algebraically closed and letting k ∈ {0, . . . , c}, we have:

deg c-k,k P X = length(H k x ∩ X ∩ H c-k y )
where H k x = V(l 1,0 , . . . , l k,0 ) is the zero locus of k general linear forms l 1,0 , . . . , l k,0 in the x-variables (that is deg(l i,0 ) = (1, 0) for all i ∈ {0, . . . , k}) and H k y =

V(l 0,1 , . . . , l 0,c-k ) is the zero locus of ck general linear forms l 0,1 , . . . , l 0,c-k in the y-variables.

Definition 1.3 (projective degrees of a map). Let f :

P n k P n k , regular on a dense open subset U ⊂ P n k , and let Γ f = { x, f (x) , x ∈ U } ⊂ P n k × P n k be the graph of f . It is an n-fold so, for k ∈ {0, . . . , n}, the k-th projective degree of f , written d k (f ), is defined by d k (f ) := deg n-k,k P Γ f .
In the following, our convention is also that

d k (f ) = 0 if k > n.
By Remark 1.2, note that d 0 (f ) = 1 if and only if f is an isomorphism between two dense open subsets of P n k in which case f is said to be a Cremona map. Following for instance [EH00, III.3.6], a bigraded free resolution of S/ I Γ f provides the numerator Num S/ I f (T 0 , T 1 ) of the Hilbert series of the coordinate ring S/ I Γ f of Γ f and thus provides the projective degrees of f . Now let us clarify in which embedding we consider the graphs Γ g and Γ g ′ of, respectively, a map g :

P m k P m k and g ′ : P m ′ k P m ′ k see Notation 2. Denote by S m = R m [y 0 , . . . , y m ], S m ′ = R m ′ [y m , . . . , y m+m ′ ], S = R[y 0 , . . . , y m+m ′ ] and I Γg ⊂ S (resp. I Γ g ′ ⊂ S) be the ideal of the graph Γ g ⊂ P m+m ′ k × P m+m ′ k of g (resp. Γ g ′ ⊂ P m+m ′ k × P m+m ′ k of g ′ ) where Γ g ⊂ P m+m ′ k × P m+m ′ k = BiProj(S). Geometrically, Γ g (resp. Γ g ′ ) is thus the cone of vertex Γ g ∩ P m k × P m k (resp. Γ g ′ ∩ P m ′ k × P m ′ k ) where P m k × P m k = BiProj(S m ) (resp. P m ′ k × P m ′ k = BiProj(S m ′ ). Proposition 1 follows from the following results about Γ g ∩ Γ g ′ .
Lemma 1.4. Assume that the tensor product F g ⊗ F g ′ , of a bigraded free resolution F g of Γ g with a bigraded free resolution F g ′ of Γ g ′ , is a free resolution of Γ g ∩ Γ g ′ . Then for any k ∈ {0, . . . , m + m ′ }:

d k ([g|g ′ ]) = k p=0 d p (g)d k-p (g ′ ).
Proof. By assumptions, the numerator Num S/ IΓ g ∩Γ g ′ (T 0 , T 1 ) of the Hilbert series of S/ I Γg ∩Γ g ′ is the product Num g (T 0 , T 1 ) Num g ′ (T 0 , T 1 ) of the numerators of the Hilbert series of respectively S/ I g and S/ I g ′ .

Focusing on the homogeneous component of total degree m + m ′ of Num [g|g ′ ] (1 -T 0 , 1 -T 1 ), we thus have:

N um S/ IΓ g ∩Γ g ′ (1 -T 0 ,1 -T 1 ) m+m ′ = Num g (1 -T 0 , 1 -T 1 ) Num g ′ (1 -T 0 , 1 -T 1 ) m+m ′ = Num g (1 -T 0 , 1 -T 1 ) m Num g ′ (1 -T 0 , 1 -T 1 ) m ′ the last equality holds because Num g (1 -T 0 , 1 -T 1 ) m (resp. Num g ′ (1 -T 0 , 1 - T 1 ) m ′ ) is also the homogeneous component of smallest total degree of Num g (1 - T 0 , 1 -T 1 ) (resp. Num ′ g (1 -T 0 , 1 -T 1 )) as Γ g , being the cone over the graph of g, is irreducible of codimension m (resp. Γ g ′ is irreducible of codimension m ′ ).
Given these preliminaries and before proving Proposition 1, let us briefly underline why, when building a glued map [g|g ′ ], we restrict to inputted determinantal maps g and g ′ . Definition 1.5 (Hilbert-Burch matrix of a determinantal map). Given that the base ideal I f = (f 0 : . . .

: f n ) ⊂ R of f verifies codim V(I f )
2 and that I f is the n-minors ideal of a (n + 1) × n-matrix, Hilbert-Burch theorem [Eis95, Theorem 20.15] states that I f is the n-minors ideal of its presentation matrix Φ f , i.e. the presentation of I f reads:

0 R n R n+1 I f 0. Φ f (f0 ... fn)
It motives why the matrix Φ f is called the Hilbert-Burch matrix of f in the following, see for instance [START_REF] Kustin | Rational normal scrolls and the defining equations of Rees algebras[END_REF], [START_REF] Kustin | The bi-graded structure of Symmetric Algebras with applications to Rees rings[END_REF] and [START_REF] Kim | Equations defining certain graphs[END_REF] for recent works about maps defined by Hilbert-Burch matrices.

When gluing two determinantal maps g and g ′ as in Proposition 1, the (m + m ′ )minors ideal of the concatenated matrix Φ [g|g ′ ] always defines a map

P m+m ′ k P m+m ′ k and, if codim V I m+m ′ (Φ [g|g ′ ] ) = 2, the base ideal I [g|g ′ ] of [g|g ′ ] is equal to I m+m ′ (Φ [g|g ′ ]
). Let us mention here that a glued map [g|g ′ ] can also be defined when g and g ′ are not necessarily determinantal however in this case [g|g ′ ] is not necessarily a map from P m+m ′ k to P m+m ′ k . Now recall that, algebraically, the graph Γ f of a map f = (f 0 : . . . : f n ) :

P n k P n k is the Proj of the Rees algebra R(I f ) = ⊕ k 0 I k f t k of the base ideal I f of f and the embedding of Γ f in P n k P n
k is defined by the surjection S ։ R(I f ) sending the variables y i to f i t (S = R[y 0 , . . . , y n ]). In practice, the kernel J of the latter surjection may be difficult to compute and an approximation of J is more accessible by considering the symmetric algebra of I f , this is a classical approach that we now summarize briefly in our context of rational maps, see [Vas05] for an introduction in a broader context and pointers to references about this procedure. The presentation matrix Φ f of I f , by definition verifying the following short exact sequence:

R n ′ R n+1 I f 0, Φ f (f0 ... fn)
defines an embedding of P(I f ) = Proj Sym(I f ) in P n k × P n k whose ideal J 1 is generated by the n ′ entries of the line matrix (y 0 . . . y n )Φ f (here we consider that Φ f is a matrix both in R and S). Since the natural surjection Sym(I f ) ։ R(I f ) factorizing the maps I ⊗k f ։ I k f defines an embedding of Γ f in P(I f ), one has that J 1 ⊂ J, i.e. J 1 provides some equations of Γ f . The ideal I f is said to be of linear type when J 1 = J.

When [START_REF] Micali | Sur les algèbre universelles[END_REF] and are described by the Fitting ideals of I f , i.e. by the ideals I k (Φ f ) for k ∈ {1, . . . , n -1} of k-minors ideals of the presentation matrix Φ f of I f , see [Vas05, Prop. 1.1 and below] and [START_REF] Busé | Torsion of the symmetric algebra and implicitization[END_REF]. Hence, the irreducible components of P(I f ) are the graph Γ f of f and eventual additional pieces lying above closed strict subschemes of the source space P n k of f . With all these facts, let us now show Proposition 1.

f is determinantal of Hilbert-Burch matrix Φ f ∈ R (n+1)×n , the kernel of Sym(I f ) ։ R(I f ) is the R-torsion of Sym(I f )
Proof of Proposition 1. Assume that codim V I m+m ′ (Φ [g|g ′ ] ) = 2 so the base ideal I [g|g ′ ] of the glued map [g|g ′ ] : P m+m ′ k P m+m ′ k is the (m + m ′ )-minors ideal of the matrix Φ [g|g ′ ] . It is clear that Γ [g|g ′ ] is included in Γ g ∩ Γ g ′ . Indeed, by definition, Γ [g|g ′ ] ⊂ P(I [g|g ′ ] ) = P(I g ) ∩ P(I g ′ ) so Γ [g|g ′ ] ⊂ P(I g ) (resp. Γ [g|g ′ ] ⊂ P(I g ′ ))
and since Γ [g|g ′ ] cannot be included in a component of P(I g ) (resp. P(I g ′ )) lying above a closed strict subscheme of

P m+m ′ k , it is necessarily included in Γ g (resp. Γ g ′ ). Now, since, by assumption, F g ⊗F g ′ is a resolution of Γ [g|g ′ ] and Γ [g|g ′ ] ⊂ Γ g ∩Γ g ′ , one has Γ [g|g ′ ] = Γ g ∩ Γ g ′ .
The conclusion of Proposition 1 follows then from Lemma 1.4.

Note that, as set, Γ [g|g ′ ] and Γ g ∩ Γ g ′ are not always equal as illustrated by the following example.

Example 1.6. Let Φ g =   x 2 0 x 1 x 0 x 2 0 x 2 1   and Φ g ′ =   x 3 0 x 4 x 2 x 3 0 x 2 4 
 and Φ [g|g ′ ] be the concatenate matrix of Φ g and Φ g ′ as in Proposition 1. Then, as it can be checked by direct computation on a computer algebra system such as Macaulay2, one has that codim

V Φ [g|g ′ ] ) 2 and that d([g|g ′ ]) = (1, 6, 10, 6, 1), d(g) = d(g ′ ) =
(1, 3, 1) and

d 2 ([g|g ′ ]) = 10 < 11 = d 0 (g)d 2 (g ′ ) + d 1 (g)d 1 (g ′ ) + d 2 (g)d 0 (g ′ ).
In this example

F g ⊗ F g ′ provides a bigraded free resolution of Γ g ∩ Γ g ′ and Γ [g|g ′ ] Γ g ∩ Γ g ′ .
Let us also mention that there exist situations where the ideal of Γ g ∩ Γ g ′ decomposes as the intersection of the ideal of Γ [g|g ′ ] and the ideal of embedded components, so that Γ [g|g ′ ] and Γ g ∩ Γ g ′ are not scheme-theoretically equal but have the same projective degrees.

Even if it might seem a bit artificial (since the results about the standard Cremona maps are well known, in particular its projective degrees, and computable by other means, see [START_REF] Gonzalez-Sprinberg | On characteristic classes of determinantal Cremona transformations[END_REF]), we illustrate an application of Proposition 1 in the example of the standard Cremona maps

τ m+m ′ = [τ m |τ m ′ ].
Proposition 1.7. Following Notation 2, let

τ m = (x 1 • • • x m : . . . : x 0 • • • x m-1 ) : P m k P m k τ m ′ = (x m+1 • • • x m+m ′ : . . . : x m • • • x m+m ′ -1 ) : P m ′ k P m ′ k with associated Hilbert-Burch matrices Φ τm ∈ R (m+1)×m m and Φ τ m ′ ∈ R (m ′ +1)×m ′ m ′ . Let also Φ [τm|τ m ′ ] ∈ R (m+m ′ +1)×(m+m ′ ) be as in Proposition 1. Then: (1) codim V(I m+m ′ Φ [τm|τ m ′ ] ) = 2, (2) the ideal of P(I [τm|τ m ′ ]
) is minimally resolved by F τm ⊗ F τ m ′ which is the Koszul complex on the entries of the line matrix y 0 . . .

y m+m ′ Φ [τm|τ m ′ ] . (3) P(I [τm|τ m ′ ] ) = Γ [τm|τ m ′ ] and, consequently, for any k ∈ {0, . . . , m + m ′ } (1.1) d k (τ m+m ′ ) = k p=0 d p (τ m )d k-p (τ m ′ ) = m + m ′ k .
Proof. Let m, m ′ 1 and

(1.2) Φ τ m+m ′ =           x0 0 . . . . . . 0 -x1 x1
x m+m ′ -1 0 . . . . . . 0 -x m+m ′ .           Since I m+m ′ (Φ [τm|τ m ′ ] ) = (x 1 • • • x m+m ′ , . . . , x 0 • • • x m+m ′ -1 ), one has that codim V I m+m ′ (Φ [τm|τ m ′ ] ) =
Γ τm ∩ Γ τ m ′ = P(I τm ) ∩ P(I τ m ′ ) = P(I [τm|τ m ′ ] ) = Γ [τm|τ m ′ ]
and Equation (1.1) follows then from a direct application of Proposition 1.

• Initial case: Item (2) and Item (3) are verified in the case m = 1, this follows from the fact that P(I τ1 ) = V(x 0 y 0x 1 y 1 ) whose minimal free resolution is the Koszul complex on the single irreducible polynomial x 0 y 0 - To show this last condition, i.e. to show that

x 1 y 1 ∈ k[x 0 ,
I m ∩ I m ′ = I m • I m ′ , we compute I m ∩ I m ′ by eliminating t in the ideal tI m + (1 -t)I m ′ ⊂ S[t]
see [AL94, Prop. 2.3.5] for this standard use of Gröbner bases. Using the graded reverse lexicographic order, the variable t being bigger than the x and y variables, and denoting ∀i ∈ {0, . . . , m -1}, g i = t(x i y ix i+1 y i+1 ), ∀j ∈ {m, . . . , m + m ′ }, g j = (1t)(x j y jx j+1 y j+1 ).

and G = {g 0 , . . . , g m+m ′ }, a direct computation shows that for any i ∈ {0, . . . , m -1} and j ∈ {m, . . . , m + m ′ } one has that S(g i , g j ) reduces to (x i y ix i+1 y i+1 )(x j y jx j+1 y j+1 ) ∈ I m • I m ′ modulo g 0 , . . . , g m+m ′ (where S(g i , g j ) is the S-polynomial associated to g i and g j ). Since S(g i , g i ′ ) reduces to 0 modulo g 0 , . . . , g m+m ′ if i, i ′ are both elements of {0, . . . , m-1} or both elements {m, . . . , m + m ′ } and since S(g i , S(g i ′ , g j )) reduces to 0 modulo g 0 , . . . , g m+m ′ , S(g 0 , g m ), . . . , S(g m-1 , g m+m ′ -1 ), one has then that

(tI m + (1 -t)I m ′ ) ∩ S ⊂ I m • I m ′
which concludes the first step and shows Item (2). We shows that the ideal I m+m ′ of P(I [τm|τ m ′ ] ) (which is generated by the entries y 0 . . . y m+m ′ Φ [τm|τ m ′ ] ) is prime again by computing Gröbner basis. More precisely, a direct computation shows that the set H = {h 0 = x 0 y 0x 1 y 1 , . . . , h m+m ′ = x m+m ′ -1 y m+m ′ -1x m+m ′ y m+m ′ } is a Gröbner basis of I m+m ′ (all S-polynomials in the h i reduces to 0 modulo h 0 , . . . , h m+m ′ -1 ). We show that I m+m ′ is prime by applying the primality test

[AL94, Algoritm 4.4.1] since, given i ∈ {0 . . . , m + m ′ -1}, h i x i y i - x i+1 y i+1 is irreducible in k ′ [x i ]
where k ′ is the quotient field of the ring k[x i+1 , . . . , x n , y 0 , . . . , y n ]/(h i+1 , . . . , h m+m ′ -1 ). The last equality (1.1) follows then from applying classical formulas between binomial numbers.

Let us emphasize again that all the previous results are well known and could be summed up by the fact that the base ideal of the standard Cremona maps is of linear type (see for instance [RS01, Subsection 2.1]). However the scheme of our proof of Proposition 1.7, could virtually be applied to more general situations. Even if it is at the moment out of our reach, let us present the kind of situations we have in mind and that we verified experimentally in a all the examples we considered.

Conjecture 1.8. Following Notation 2 about glued maps, assume moreover that k is algebraically closed, and let

                Φ g 0 m×m ′ 0 m ′ ×m Φ g ′ m m ′ + 1 m m ′ m + 1 m ′ Φ [g|g ′ ] = (Φ ij ) 0 i m+m ′ +1 1 j m+m ′ = ∈ R (m+m ′ +1)×(m+m ′ )
be the matrix defined by the following data:

• for any j ∈ {1, . . . , m}, let k j 2, λ 1,j , . . . , λ kj ,j ∈ R m = k[x 0 , . . . , x m ] ⊂ R and:

for i ∈ {1, . . . , m + 1}, let Φ ij ∈ |λ 1,j , . . . , λ kj ,j | be a general linear combination of λ 1,j , . . . , λ kj ,j , -for i ∈ {m + 2, . . . , m + m ′ + 1}, Φ ij = 0. • for any j ∈ {m + 1, . . . , m + m ′ }, let k j 2 and λ 1,j , . . . , λ kj ,j ∈ R m ′ = k[x m , . . . , x m+m ′ ] ⊂ R and:

for i ∈ {1, . . . , m}, Φ ij = 0, -for i ∈ {m + 1, . . . , m + m ′ + 1}, let Φ ij ∈ |λ 1,j , . . . , λ kj ,j | be a general linear combination of λ 1,j , . . . , λ kj ,j .

Let also Φ g = (Φ ij ) 0 i m+1

1 j m ∈ R (m+1)×m m and Φ g ′ = (Φ ij ) m+1 i m+m ′ +1 m+1 j m+m ′ ∈ R (m ′ +1)×m ′ n
and [g|g ′ ] : P m+m ′ P m+m ′ , g : P m P m and g ′ : P m ′ P m ′ be the determinantal maps defined by Φ [g|g ′ ] , Φ g and Φ g ′ (where

P m = Proj(R m ) ⊂ Proj(R) = P m+m ′ and P m ′ = Proj(R m ′ ) ⊂ Proj(R) = P m+m ′ ).
Then for all i ∈ {0, . . . , m + m ′ }:

(1.3) d k ([g|g ′ ]) = k p=0 d p (g)d k-p (g ′ ).

Projective degrees vs mixed volumes of Koszul-determinantal maps

For the rest of the paper, we let k = C. Given a map f : P n k P n k of base ideal I f , the inclusion Γ f ⊂ P(I f ) of the graph Γ f of f in the Proj of the symmetric algebra of I f , see below Definition 1.5 for more details, provides the following estimations:

(2.1) ∀k ∈ {0, . . . , n}, d k (f ) deg n-k,k P P(I f ).
Let us now present a specificity when computing the projective degrees of the Koszul-determinantal map f we are now going to define. Recall Definition 1.5 that, assuming that f is determinantal, the presentation matrix Φ f of I f is called the Hilbert-Burch matrix of f . In this latter case, letting

0 n ⊕ k=1 R(-d k ) R n+1 I f (d) 0 Φ f (f0 ... fn)
be a graded free resolution of the base ideal of f , we call the positive integer d 1 , . . . , d n 1 the Hilbert-Burch degrees of f . Remark from the equations (y 0 . . . y n )Φ f defining the embedding of P(I f ) in

P n k × P n k that the conditions: (2.2) ∀k ∈ {1, . . . , n -1}, codim V I k (Φ f ) ≥ n + 1 -k
are the required conditions in order that P(I f ) has codimension n in which case P(I f ) is a complete intersection and a minimal free resolution of its coordinate ring is the Koszul complex on the entries of (y 0 . . . y n )Φ f . Definition 2.1 (Koszul-determinantal map). A Koszul-determinantal map f = (f 0 : . . . : f n ) : P n k P n k is a map such that the following two conditions are verified:

• a minimal presentation of

I f = (f 0 , . . . , f n ) reads: 0 R n R n+1 I f 0. Φ f (f0 ... fn)
• the conditions (2.2) are satisfied.

Proposition 2.2. Let f = (f 0 : . . . : f n ) : P n k P n k be a Koszul-determinantal map of Hilbert-Burch degree d 1 , . . . , d n 1, then for all k ∈ {0, . . . , n},

(2.3) d k (f ) σ n-k,n (d 1 , . . . , d n ) = {i1,...,i k }⊂{1,...,n} d i1 . . . d i k
where σ k,n (u 1 , . . . , u n ) is the k-th symmetric polynomial in n variables.

Proof. Letting Φ f be the (n+1)×n Hilbert-Burch matrix of f , the ideal J 1 of P(I f ) in P n k × P n k is generated by the n entries φ 1 , . . . , φ n of the line matrix (y 0 . . . y n )Φ f . Since conditions (2.2) are moreover satisfied, P(I f ) has codimension n in P n k × P n k so P(I f ) is a complete intersection and the Koszul complex on φ 1 , . . . , φ n ∈ S 0 S(-

n k=1 d k , -n) . . . 2 ∧ n ⊕ k=1 S(-d k , -1) n ⊕ k=1 S(-d k , -1) S
provides a minimal bi-graded free resolution of the coordinate ring of P(I f ) from which we extract the component of total degree n of Num S/ J1 (1

-T 0 , 1 -T 1 ) (see [EH00, III.3.6]): ∀k ∈ {0, . . . , n}, deg n-k,k P P(I f ) = s n-k,n (d 1 , . . . , d n ) = {i1,...,i k }⊂{1,...,n} d i1 . . . d i k .
The conclusion of the proposition follows then from (2.1).

We point out that the conclusion of Proposition 2.2 is classical and had been established in more general context of maps f : P n k P n ′ k with n n ′ , see for instance [START_REF] Cid-Ruiz | Mixed multiplicities and projective degrees of rational maps[END_REF]Theorem 5.7]. It explains however why, when interested in determinantal Cremona maps, one has also to consider base ideal not of linear type or else, if the base ideal is of linear type, the entries of the presentation matrix can only contain linear polynomials (case d 1 = . . . = d n = 1 of the previous proposition).

In other words, one has to consider the kernel of the surjection Sym(I f ) ։ R(I f ) in order to understand the level of approximations of the inequalities (2.3). As we previously explained in Section 1, this kernel is described by the ideals I k (Φ f ) for k ∈ {1, . . . , n -1} of k-minors ideals of the presentation matrix Φ f of I f which, in our context of determinantal base ideal I f , have an expected codimension:

∀k ∈ {1, . . . , n}, codim V I k (Φ f ) ≥ n + 2 -k,
ensuring, if these latter conditions are verified, that I f is of linear type [RS01, Subsection 2.1]. Hence the conditions (2.2) are the first step to consider when interested in determinantal Cremona maps.

2.1. Bernstein theorem's bound on the number of solutions of a zero dimensional polynomial system. When interested in the topological degree d 0 (f ) of a determinantal map f = (f 0 : . . . : f n ) : P n k P n k , remark that two polynomial systems can be considered. An initial one, in the end the main object of this work, is defined by f itself: given y = (y 0 : . . . : y n ) ∈ P n k in the target space of f , one wants to find the pre-images x ∈ P n k of y, i.e. wants to solve the polynomial system (2.4)

f (x) = y ⇔        y 0 = f 0 (x 0 , . . . : x n ) . . . y n = f n (x 0 , . . . : x n )
Expressing y as the intersection of n hyperplans in the target space of f and assuming that y is general enough, Bézout theorem then asserts that d 0 (f ) = #f -1 ({y}) d n where d = d n-1 (f ) is the common degree of the polynomials f i generating the base ideal I f = (f 0 , . . . , f n ) of f . An intermediate polynomial system is moreover defined by a presentation matrix Φ f of I f . Indeed, by definition of Φ f , one has (f 0 . . . f n )Φ f = 0 so the polynomial system (2.5) (y 0 . . .

y n )Φ f = 0
whose number of equations is the number of columns of Φ f , contains by definition f -1 ({y}). Now consider the situation where Φ f = (φ ij ) 0 i n

1 j n
is a (n + 1) × n matrix where each entry φ ij of the j-th column of Φ f has degree d j 1 and in which case

d = d n-1 (f ) = d 1 + . . . + d n by Hilbert-Burch theorem. Then, provided that codim V I k (Φ f ) ≥ n + 1 -k for all k ∈ {1, . . . , n}, the system (2.5) is 0-dimensional in P n
k and, by Bézout theorem, has d 1 • . . . • d n solutions. Moreover, in case entries of Φ f are sparse polynomials that all verify the same algebraic constraints on their coefficients, one can refine the bound on the number of solutions of (2.5) by using Bernstein's theorem on sparse polynomials. It then gives a combinatorial translation, via the computation of mixed volumes associated to the polynomial entries of Φ f , to the problem of detecting determinantal Cremona maps of given Hilbert-Burch degree (d 1 , . . . , d n ) among all determinantal maps of Hilbert-Burch degree (d 1 , . . . , d n ).

This idea structures our approach of Proposition 2.14, see [START_REF] Trung | Mixed multiplicities of ideals versus mixed volumes of polytopes[END_REF] or [START_REF] Lazarsfeld | Convex bodies associated to linear series[END_REF] for developments in more general contexts and for pointers to the related literature. We refer to [CLO05, Chapter 7] for the background in convex geometry and all the material we use for the actual computations. Following [CLO05, 7.4], a set C ⊂ R n is convex if it contains any segments between two points in C and the convex hull Conv(S) of a subset S ⊂ R n is the smallest convex set containing S. A polytope is the convex hull Conv(A) of a finite set A ⊂ R n and the polytopes which are the convex hull of points with integer coordinates are called lattice polytopes.

Given φ =

α∈N n c α x α ∈ k[x 1 , . . . ,
x n ] where x α = x α1 1 . . . x αn n , the Newton polytope of φ, denoted NP(φ), is the lattice polytope NP(φ) = Conv{α ∈ N n , c α = 0}.

A polytope P ⊂ R n has an n-dimensional volume Vol n (P ). Given two polytopes P, Q ⊂ R n and a real number λ 0, the Minkowski sum of P and Q, denoted P + Q is the set P + Q := {p + q, p ∈ P, q ∈ Q} where p + q denotes the usual vector sum in R n and the λP stands for the polytope {λp, p ∈ P } where λp is the usual scalar multiplication in R n . Given any collection P 1 , . . . , P r ⊂ R n and r non negative scalar λ 1 , . . . , λ r ∈ R, then Vol n (λ 1 P 1 + . . . λ r P r ) is a homogeneous polynomial of degree n in the λ i [CLO05, 7. Prop.4.9]. Definition 2.3 (mixed volume of a collection of polytopes). The n-dimensional mixed volume MV n (P 1 , . . . , P n ) of given polytopes P 1 , . . . , P n is the coefficient of the monomial λ 1 , . . . , λ n in Vol n (P 1 , . . . , P n ). where π n ′ : R n+n ′ → R n ′ stands for the projection on the last n ′ coordinates.

Following [CLO05, 7 Section 5], let us now define the genericity of a polynomial with respect to a polytope.

Definition 2.6 (genericity with respect to a polytope). Given finite set

A ⊂ Z n , put L(A) := { α∈Ai c α x α ∈ k[x 1 , . . . ,
x n ]} and remark that each L(A) can be considered as an affine space k #Ai with the coordinate c α as coordinates.

A polynomial α∈Ai c α x α is said to be generic with respect to L(A) if its coefficients are generic in L(A).

Given k 1 and finite sets A 1 , . . . , A k ∈ Z n , a property is said to hold generically for polynomials (φ 1 , . . . , φ n ) ∈ L(A 1 )× . . .× L(A k ) is there is a non zero polynomial in the coefficients of the φ i such that the property holds for all φ 1 , . . . , φ n for which the polynomial is non vanishing, in particular if every φ 1 , . . . , φ n is generic with respect to its own polytope L(A 1 ), . . . , L(A k ).

Theorem 2.7 (Bernstein theorem). Given polynomials φ 1 , . . . , φ n ∈ k[x 1 , . . . , x n ] with finitely many common zeroes in (k * ) n , let P i = NP(φ i ). Then the number of common zeroes in (C * ) n is bounded above by the mixed volume MV n (P 1 , . . . , P n ). Moreover if each φ i is generic with respect to P i , the number of common zero solutions is exactly MV n (P 1 , . . . , P n ).

See [CLO05, Proof of 7. Th. 5.4] and the references therein for highlights about Bernstein theorem.

2.2. Projective degrees of Koszul-determinantal maps defined by sparse polynomials. Given a homogeneous polynomial φ ∈ R = k[x 0 , . . . , x n ], the Newton polytope NP(φ) ⊂ R n associated to φ is the Newton polytope of the dehomogenization of φ with respect to a variable x i (omitted when irrelevant) and, reciprocally, given a polytope P ⊂ R n , we denote φ ∈ P for a homogeneous polynomial φ ∈ R whose de-homogenization with respect to the variable x i is in P . Furthermore, we consider more general Newton polytopes P ⊂ R n × R n corresponding to bi-homogeneous polynomials φ ∈ S = R[y 0 , . . . , y n ], that is the bi-dehomogenization of φ with respect to one fixed variable x i and one fixed variable y j (i, j ∈ {0, . . . , n}) is in P .

Lemma 2.10. Let m, m ′ ∈ N * and:

•

P 1 , . . . , P m ⊂ R m × {0 R m ′ } ⊂ R m × R m ′ , • P m+1 , . . . , P m+m ′ ⊂ {0 R m } × R m ′ ⊂ R m × R m ′ , be m + m ′
polytopes. For j ∈ {1, . . . , m + m ′ } put:

P y j := P j × {0 R m+m ′ } + {0 R m+m ′ } × S y m+m ′ ⊂ R m+m ′ × R m+m ′ . Then for all k ∈ {0, . . . , m + m ′ }, MV 2(m+m ′ ) (S x
⊂ R m+m ′ × R m+m ′ (resp. S y m+m ′ ) as the sum S x m × {0 R m ′ } × {0 R m+m ′ } + {0 R m } × S x m ′ × {0 R m+m ′ } ⊂ R m+m ′ × R m+m ′ (resp. as the sum {0 R m+m ′ } × S y m × {0 R m ′ } + {0 R m+m ′ } × {0 R m } × S y m ′ × {0 R m+m ′ })
and then applying the multilinearity of the mixed volume (Theorem 2.4. Item (ii)), the projection formula (Lemma 2.5), the vanishing condition (Theorem 2.4. Item (iii)) and eventually Lemma 2.9. )

P(I f )\Γ f ⊂ V( n Π i=0 x i ) ⊂ P n k × P n k
where for l ∈ {1, . . . , n}, P y l ⊂ R n × R n is the Newton polytope of the l-th entry of the matrix (y 0 . . .

y n )Φ f . Proof. Assume that P(I f )\Γ f ⊂ V( n Π i=0 x i ) ⊂ P n k × P n k so P(I f )
k (f ) < (2.7).
Hence, to know the actual term d k (f ) of the projective degrees of a determinantal map f , the computation of the associated mixed volume has to be completed by a preliminary control on the support of the successive ideal of minors of Φ f . We illustrate such a control in the next subsection.

(g ′ ) ij ) 0 i n m+1 j m+m ′ ∈ R (m ′ +1)×m ′ m ′ and Φ [g|g ′ ] = (φ ij ) 0 i m+m ′ 1 j m+m ′ ∈ R (m+m ′ +1)×(m+m ′ ) m+m ′
where R m+m ′ = k[x 0 , . . . , x m+m ′ ] be such that:

• for any j ∈ {1, . . . , m}, each entry φ (g) ij of the j-th column of Φ g is such that φ In the following, under the notation of Proposition-Definition 2.12 and under the assumption that codim V I m (Φ g ) = codim V I m ′ (Φ g ′ ) = 2, we let g : P m k P m k (resp. g ′ : P m ′ k P m ′ k ) be the map whose base locus I g is equal to I m (Φ g ) (resp. I g ′ = I m (Φ g ′ )).

Proof. Assume that codim V I m (Φ g ) = codim V I m ′ (Φ g ′ ) = 2 and suppose by contradiction that codim V I m+m ′ (Φ [g|g ′ ] ) < 2. Then there is a common factor to each m + m ′ minors of Φ [g|g ′ ] , so, after operation on columns, one column of Φ [g|g ′ ] have all its entries sharing a common factor. But it is impossible under the genericity assumption on the entries of Φ [g|g ′ ] .

In the following example, we illustrate however that the glued map [g|g ′ ] of two Koszul-determinantal maps g and g ′ , though determinantal by Proposition-Definition 2.12, may not be Koszul-determinantal.

Example 2.13. Put m = n = 2 and let l 1 = l 2 = l 3 = l 4 = 2 with:

• ψ (1) 0 = ψ (2) 0 = x 1 , ψ (1) 1 = ψ (2) 1 = x 2 • ψ (3) 0 = ψ (4) 0 = x 2 , ψ (3)

  2 (the generators of I m+m ′ (Φ [τm|τ m ′ ] ) do not share a common factor) and thus Item (1) is verified. The proof of Item (2) and Item (3) relies on an induction on m and m ′ . The inductive step consists in first showing that the tensor product F τm ⊗ F τ m ′ of a free resolution F τm of P(I τm ) and a free resolution F τ m ′ of P(I τ m ′ ) provides a free resolution of P(I τm ) ∩ P(I τ m ′ ) = P(I [τm|τ m ′ ] ). The second step consists in showing that P(I [τm|τ m ′ ] ) is irreducible or, in other words, that I P(Iτ m ) + I P(Iτ m ′ ) is prime. This latter property insures that

  x 1 , y 0 , y 1 ]. • Inductive step: let m, m ′ 1 and assume that Item (2) and Item (3) hold for τ m and τ m ′ . In particular the ideal I m of P(I τm ) = Γ τm (resp. the ideal I m ′ of P(I τ m ′ ) = Γ τ m ′ ) is minimally resolved by the Koszul complex F τm on the entries of the line matrix y 0 . . . y m Φ τm (resp. by the Koszul complex F τ m ′ on the entries of the line matrix y m . . . y m ′ Φ τ m ′ ). We show Item (2) and Item (3) via classical methods involving Gröbner bases and we refer to [AL94] for the associated definitions associated to this argument. Remark that F τm ⊗ F τ m+m ′ is the Koszul complex on the entries of y 0 . . . y m+m ′ Φ [τm|τ m ′ ] and one has that F τm ⊗ F τ m ′ resolves the ideal of P(I [τm|τ m ′ ] ) if and only if ∀i 1, Tor i S/I m , S/I m ′ = 0, S being the polynomial ring k[x 0 , . . . , x m+m ′ , y 0 , . . . , y m+m ′ ]. Actually, by Tor rigidity, the latter conditions are verified if and only if Tor 1 S/I m , S/I m ′ = I m ∩ I m ′ /I m • I m ′ = 0.

Proposition 2. 11 .

 11 Let f : P n k P n k be a Koszul-determinantal map of Hilbert-Burch matrix Φ f = (φ ij ) 0 i n 1 j n . Then: ∀k ∈ {0, . . . , n}, d k (f ) = MV 2n (S x n , . . . , S x n k , P y 1 , . . . , P y n , S y n , . . . , S y n n-k

2. 3 .

 3 Gluing determinantal maps subject to general conditions. Let us now focus on another construction of a glued determinantal map [g|g ′ ] : P m+m ′ k P m+m ′ k , where m, m ′ ∈ N * , starting from two determinantal Cremona maps g : P m k P m k and g ′ : P m ′ k P m ′ k . In the following, given polynomials φ, ψ 1 , . . . , ψ l , we denote by φ ∈ |ψ 1 , . . . , ψ l | the condition that φ = l k=1 λ k ψ k is a general linear combination of ψ 1 , . . . , ψ l .Proposition-Definition 2.12 (glued map). Given any j ∈ {1, . . . , m + m ′ }, let l j ∈ N and let:• ψ (j) 0 , . . . , ψ (j) lj ∈ R m = k[x 0 , . . . , x m ] if j ∈ {1, . . . , m} • ψ (j) 0 , . . . , ψ (j) lj ∈ R m ′ = k[x m , . . . , x m+m ′ ] if j ∈ {m + 1, . . . , m + m ′ }. Let also Φ g = (φ m , Φ g ′ = (φ

  • for any j ∈ {1, . . . , m ′ }, each entry φ(g ′ ) ij of the j-th column of Φ g ′ is such that φ lj +m |. • for any j ∈ {1, . . . , m + m ′ }, each entry φ ij of the j-th column of Φ [g|g ′ ] is such that φ ij ∈ |ψ Assume that codim V I m (Φ g ) = codim V I m ′ (Φ g ′ ) = 2, then codim V I m+m ′ (Φ [g|g ′ ] ) = 2and define the glued map [g|g ′ ] : P m+m ′ k P m+m ′ k as the map whose base locus I [g|g ′ ] is the (m + m ′ )-minors ideal of Φ [g|g ′ ] .

  that codim V I 1 (Φ [g|g ′ ] ) = 3 < 4 (I 1 (Φ [g|g ′ ] ) = (x 1 , x 2 , x 3 )).Let us now describe the projective degrees of a glued determinantal map in the almost linear setting.Proposition 2.14. Let m, d 1, put l 1 = . . . = l m = m+1, l m+1 = 1, l m+2 = m+3• ∀j ∈ {1, . . . , m}, k ∈ {0, . . . , m}, let ψ • ∀k ∈ {1, . . . , 3d}, ψ(m+2) k is the k-th generator of the product (x m , x m+1 , x m+2 ) • (x m+1 , x m+2 ) d-1 = (x m x d-1 m+1 , x m x d-1 m+2 ) + (x m+1 , x m+2 ) d . Let [g|g ′ ] : P m+2 k P m+2k be the glued map associated to the previous data as defined in Proposition-Definition 2.12 and whose base idealI [g|g ′ ] is the (m + 2)minors ideal of Φ [g|g ′ ] .Then [g|g ′ ] is a determinantal map and moreover:∀k ∈ {0, . . . , m + 2}, d k ([g|g ′ ]) = m mk + (d + 1) m mk + 1 + m mk + 2with the convention that j i = 0 if i < 0 or i > j. Proof. First, under our general assumptions, codim I m+2 (Φ [g|g ′ ] ) = 2 and for all k ∈ {1, . . . , n}, codimI k (Φ [g|g ′ ] ) 3 + mk, hence [g|g ′ ] is Koszul-determinantal. Moreover P(I [g|g ′ ] )\Γ [g|g ′ ] ⊂ V( n Π i=0 x i ) ⊂ P m+2 k × P m+2k hence, by Proposition 2.11, we can use the mixed volumes of the polytopes P j = NP(ψ (j) 1 ) + . . . + NP(ψ (j) lj ) (j ∈ {1, . . . , m + 2}) to compute the projective degrees of [g|g ′ ].

  Theorem 2.4.[START_REF] Cox | Using Algebraic Geometry[END_REF] 7. Th.4.12] (i) The mixed volume MV n (P 1 , . . . , P n ) is invariant if the P i are replaced by their images under a volume-preserving transformation of R n .(ii) MV n (P 1 , . . . , P n ) is symmetric and linear in each variable (multilinearity of the mixed volume). (iii) MV n (P 1 , . . . , P n ) 0 and MV n (P 1 , . . . , P n ) = 0 if one of the P i has dimension zero (i.e. if P i consists of a single point).MV n (P 1 , . . . , P n ) > 0 if every P i has dimension n. Let n, n ′ ∈ N * and let P 1 , . . . , P n be polytopes in R n+n ′ and P n+1 , . . . , P n+n ′ be polytopes in R n × {0 R n ′ } ⊂ R n+n ′ . Then:MV n+n ′ (P 1 , . . . , P n+n ′ ) = MV n (P 1 , . . . , P n ) MV n ′ (π n (P n+1 ), . . . , π n ′ (P n+n ′ ))

	(iv) MV n (P 1 , . . . , P n ) =	n	(-1) n-k	Vol n (	P i ) where	P i is the Min-
		k=1	I⊂{1,...,n} |I|=k	i∈I	i∈I
	kowski sum of polytope.
	In addition to Theorem 2.4, our work relies also on the following toolbox.
	Lemma 2.5. [ST10, Lemma 6]

  m+m ′ , . . . , S x MV m ′ (S m ′ , . . . , S m ′

	m+m ′	, P y 1 , . . . , P y m+m ′ , S y m+m ′ , . . . , S y m+m ′	)
	m+m ′ -k	k
		=
	k
	p=0 {l 1 ,...,lp}⊂ {1,...,m ′ }

m ′ -p , P l1 , . . . , P lp ) × {l 1 ,...,l k-p }⊂ {m ′ +1,...,m+m ′ } MV m (S m , . . . , S m m+p-k , P l1 , . . . , P l k-p ) Proof. The formula follows from by decomposing first S x m+m ′

  and Γ f coincide away the coordinate exis defined by the x i . Now for any k ∈ {0, . . . , n}, is the zero locus of ck general linear forms in the y-variables. So (2.7) is also equal to the number of point with non zero coordinate in the x-variable of the reduced scheme H k linear subspace, this last quantity is equal to d k (f ) which show the indirect implication in Proposition 2.11. ). Hence, there is a k ∈ {0, . . . , n} such that for anyH kx which is the zero locus of k general linear forms in the x-variables and any H k y which is the zero locus of ck general linear forms in the y-variables, the scheme H kx ∩ T ∩ H n-k y contain a point with all non zero coordinate. Since this point is taken in account in the mixed volume (2.7), one has the strict inequality d

	Now assume that P(I f )\Γ f ⊂ V( i=0 n Π	x i ). This implies that there exists a codi-
	mension n irreducible component T of P(I f ), distinct from Γ f , that is not included in V( n Π i=0 x i ). Since T lies above the zero locus of some Fitting ideals of Φ f , it not
	included neither in V(
	(2.7)	M V 2n (S x n , . . . , S x n	, P y 1 , . . . , P y n , S y n , . . . , S y n	)
			k		n-k
	is the number of point with non zero coordinate in the x-variable of the reduced
	scheme H k x ∩ P(I f ) ∩ H n-k y	) where H k x is the zero locus of k general linear forms
	in the x-variables and H n-k y

x ∩ Γ f ∩ H n-k y ). But since H k x and H n-k y are general n Π i=0 y i

Let us also denote by S n = Conv{(0, . . . , 0 n ), (1, 0, . . . , 0 n-1 ), . . . , (0, . . . , 0 n-1 , 1)} ⊂ R n the unit simplex of R n and let S x n := S n ×{0 R n } ⊂ R n ×R n and S y n :

Proposition 2.8. Let f : P n k → P n k be a determinantal map of Hilbert-Burch matrix

Then for all k ∈ {0, . . . , n}:

where for l ∈ {1, . . . , n}, P y l ⊂ P n k × P n k is the Newton polytope of the l-th entry of the matrix (y 0 . . . y n )Φ f and σ k,n is the k-th symmetric polynomial in n-variables.

Proof. As explained in Remark 1.2, for all k ∈ {0, . . . , n}, deg n-k,k P P(I f ) is the number of common zero solutions in ⊂ P n k × P n k of the polynomials (2.6) l 1,0 , . . . , l k,0 , φ 1 , . . . , φ n , l 0,1 , . . . , l 0,n-k

where l 1,0 , . . . , l k,0 (resp. l 0,1 , . . . , l 0,n-k ) are generic with respect to S x n (resp. generic with respect to S y n ) and φ 1 , . . . , φ n ∈ S = R[y 0 , . . . , y n ] are the entries of the matrix (y 0 . . . y n )Φ f (the associated polynomial system being 0-dimensional by definition of a determinantal map, see Definition 1.5).

In this setting, by Theorem 2.7 in the generic case, the quantity

) is the number of solutions of (2.6) whose coordinates are all non zero which shows the right hand side inequality of Proposition 2.8 since the number of common zero of (2.6) is equal to σ n-k,n (d 1 , . . . , d n ).

The set of common zero of (2.6) contains moreover the set

where H k x = V(l 1,0 , . . . , l k,0 ) is the zero locus of l 1,0 , . . . , l k,0 , Γ f is the graph of f and H n-k y = V(l 0,1 , . . . , l 0,n-k ). However, by the genericity assumptions, remark that the points of

y have all non zero coordinates which shows the left hand side inequality of Proposition 2.8.

Let us now isolate two technical facts that rely on Lemma 2.5: Lemma 2.9. Let P 1 , . . . , P n ⊂ R n be n polytopes and for j ∈ {1, . . . , n} put P y j :=

Proof. It suffices to apply the multilinearity of the mixed volume (Theorem 2.4. Item (ii)), the projection formula (Lemma 2.5) and the vanishing condition (Theorem 2.4. Item (iii)) on the left member of the equality to obtain:

MV 2n (P

with the convention that d p (g) = 0 if p > m and d p (g ′ ) = 0 if p > 2. The result of Proposition 2.14 follows from the fact that:

• d(g) = (1, m 1 , . . . , m m-1 , m m ) as g is general determinantal map, see [GSP06, Theorem 2] • d(g ′ ) = (d 0 (g ′ ), d 1 (g ′ ), d 2 (g ′ )) = (1, d + 1, 1), this follows for instance from [BCRS20, Th. 5.14].


Marcos Eduardo Valle

Samuel Francisco

Marco Aurélio Granero

Santiago Velasco-Forero

Marco Aurélio Granero

Measuring the Irregularity of Vector-Valued Morphological Operators using Wasserstein Metric

Keywords: mathematical morphology, vector-valued images, total order, irregularity issue, optimal transportation

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Mathematical morphology (MM) is a nonlinear theory that uses geometric and topological concepts for image and signal processing. The theory of mathematical morphology is usually defined on an algebraic structure called complete lattices which is satisfactory for binary and grayscale images [START_REF] Ronse | Why Mathematical Morphology Needs Complete Lattices[END_REF][START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF]. In the case of vector-valued images, vector spaces endowed with a total order is one of the most comfortable frameworks for the extension of morphological processing [START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF][START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF][START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. Approaches that have been recently formulated using total orderings include [START_REF] Lézoray | Complete lattice learning for multivariate mathematical morphology[END_REF][START_REF] Sangalli | Color Mathematical Morphology Using a Fuzzy Color-Based Supervised Ordering[END_REF][START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection[END_REF][START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF][START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF][START_REF] Velasco-Forero | Mathematical Morphology for Vector Images Using Statistical Depth[END_REF]. Despite their successful applications for color and hyperspectral image processing, Chevallier and Angulo showed that the information contained in a total order is too weak to reproduce the natural topology of the value space [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. As a consequence, morphological operators may introduce irregularities and aliasing on images.

Our motivation is to formulate quantitative measures to study the irregularity implied by a morphological operator on vector-valued images. We believe that this is the first work proposing a framework based on the Wasserstein metric to score this irregularity effect considering pairs of input/output images [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF][START_REF] Villani | Topics in Optimal Transportation[END_REF]. The paper is organized as follows: Section 2 introduced mathematical morphology concepts for vector-valued images and the difficulties produced by operators based on total orders. Section 3 presents our proposition of irregularity measure. Additionally, results on natural images show the goodness of proposed measures. The paper ends with Section 4 including conclusions and recommendations for future works.

Basic Concepts on Mathematical Morphology

Let us begin by presenting the basic concepts and the notations used in this paper. First, an image I corresponds to a mapping from a point set D to a value set V, that is, I : D → V. The set of all images from a domain D to V is denoted by V = V D . Throughout the paper, we assume the point set D is finite and included in a space E, where (E, +) is a group. Usually, we consider E = R 2 or E = Z 2 with the usual addition. Furthermore, we assume the value set V is a complete lattice equipped with a metric d : V × V → [0, +∞). Recall that a complete lattice L is a partially ordered set in which any subset X ⊂ L has both an infimum and a supremum [START_REF] Birkhoff | Lattice Theory[END_REF]. The infimum and the supremum of X are denoted respectively by X and X.

Mathematical Morphology on Complete Lattices

Mathematical morphology (MM) is mainly concerned with image operators used to extract relevant geometric and topological information from an image [START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF][START_REF] Dougherty | Hands-on Morphological Image Processing[END_REF][START_REF] Soille | Morphological Image Analysis[END_REF]. The two elementary operators of MM are dilations and erosions. Many other operators, such as opening, closing, and the morphological gradient, are obtained by combining the elementary morphological operators.

Complete lattices provide an appropriate mathematical background for defining the elementary morphological operators [START_REF] Ronse | Why Mathematical Morphology Needs Complete Lattices[END_REF][START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF]. Indeed, the elementary morphological operators are those that commute with the supremum and the infimum operations in a complete lattice. When the value set V is a complete lattice, the operators δ S , ε S : V → V given by the following equations where S ⊆ E is finite are respectively a dilation and an erosion:

δ S (I)(p) = s∈S p-s∈D I(p -s) and ε S (I)(p) = s∈S p+s∈D I(p + s). (1)
The set S is referred to as the structuring element (SE) [START_REF] Soille | Morphological Image Analysis[END_REF]. The images δ S (I) and ε S (I) are respectively the dilation and the erosion of I by the structuring element S.

Although there exist more general definitions, the elementary morphological operators given by (1) are widely used in practical situations. Combining dilations and erosions, we obtain many other morphological operators. In this paper, we focus on elementary operators defined by [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF]. We also consider openings γ S and closings φ S , which are obtained by the compositions γ S = δ S • ε S and φ S = ε S • δ S [START_REF] Soille | Morphological Image Analysis[END_REF].

Vector-Valued Mathematical Morphology

Let us now address morphological operators for vector-valued images. A vector-valued image is obtained by considering V ⊆ Rd , where R = R ∪ {-∞, +∞} and d ≥ 2.

Vector-valued dilations and erosions can be defined using (1) whenever the vectorvalued set V is a complete lattice. However, there are many different ordering schemes for vector-valued sets. Defining an appropriate ordering scheme is one of the main challenges of vector-valued MM. The following references provide a brief sense of interesting directions of research on vector-valued MM [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF][START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF][START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF][START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF][START_REF] Van De Gronde | Group-Invariant Colour Morphology Based on Frames[END_REF][START_REF] Lézoray | Complete lattice learning for multivariate mathematical morphology[END_REF][START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection[END_REF][START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF][START_REF] Velasco-Forero | Mathematical Morphology for Vector Images Using Statistical Depth[END_REF][START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF].

As examples of ordering schemes on vector-valued sets, let us recall the marginal and the lexicographical orderings [START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF]. The marginal ordering is defined as follows for all u = (u 1 , . . . , u d ) ∈ V and v = (v 1 , . . . , v d ) ∈ V:

u ≤ M v ⇐⇒ u i ≤ v i , ∀i = 1, . . . , d, (2)
where "≤" denotes the usual ordering on R. The marginal ordering is also called the component-wise ordering or the Cartesian product ordering. The lexicographical ordering is defined as follows:

u ≤ L v ⇐⇒ ∃i : u i ≤ v i and u j = v j , ∀j < i. (3)
In contrast to the marginal ordering, the lexicographical ordering is a total ordering. Hence, either one of the inequalities u ≤ L v or v ≤ L u holds for any u, v ∈ V.

The False "Colors" Problem Versus the Irregularity Issue

One problem on vector-valued MM is the creation of "false colors" or, more generally, false values [START_REF] Serra | The "False Colour" Problem[END_REF]. A morphological operator ψ : V → V introduces false values whenever there are values on ψ(I) which do not belong to the original image I. Formally, let 2 V denote the power set of V and let V : V → 2 V be the mapping given by

V (I) = {I(p) : p ∈ D}, ∀I ∈ V. (4)
A morphological operator ψ introduces false colors if the set difference V (ψ(I)) \ V (I) is not empty. The abnormal false values can be a problem in many applications such as when dealing with satellite data [START_REF] Serra | The "False Colour" Problem[END_REF]. Using the marginal ordering, the dilation and the erosion given by (1) usually yield false colors. A total ordering, such as the lexicographical ordering, circumvents the problem of the false values [START_REF] Serra | The "False Colour" Problem[END_REF]. Using a total ordering, the supremum and the infimum of a finite set is an element of the set, i.e., they coincide with the maximum and minimum operations, respectively. As a consequence, if D is finite, the elementary morphological operators given by ( 1) only contain values of the input image I. On the downside, a total ordering can be irregular in a metric space. According to Chevallier and Angulo, the irregularities follow because the topology induced by a total order may not reproduce the topology of a metric space [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. Specifically, let the value set V be a totally ordered set as well as a metric space, with metric d : V × V → [0, +∞). Chevallier and Angulo showed that there exists u, v, w under mild conditions with respect to the connectivity of V. In words, although w is closer to u than v, the inequalities u ≤ v ≤ w suggest w is farther from u than v.

∈ V such that u ≤ v ≤ w but d(u, w) < d(u, v)
Since the morphological operators are defined using the extrema operators, they do not take the metric of V into account.

A visual interpretation of the irregularity issue is shown in Figure 1, which is very similar to an example provided in [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. Figure 1b) shows an image with three RGB colors, namely u = (0, 0, 0), v = (0, 0, 1), and w = (0.005, 0, 0). The toy image I is obtained by replacing pure black values u by w with probability 0.3 from an image of size 32 × 64 with two stripes of colors blue and black. The dilations J L and J M by a cross as the structuring element using respectively the lexicographical RGB and the marginal ordering schemes are also depicted in Figure 1. Visually, u and w are black colors while v is a pure blue. Using the Euclidean distance, we obtain d(u, v) = 1 and d(u, w) = 0.005. These distances agree with our color perception. However, using the lexicographical ordering, we obtain u ≤ L v ≤ L w. As a consequence, the following happens when we compute the dilation δ S (I) = J L using the lexicographical ordering: the blue pixel value v advances over the black u but it is overlaid by the black w, resulting in the irregularities shown in Figure 1a). In contrast, the dilated image depicted in Figure 1c) obtained using the marginal ordering does not present any visual irregularity.

Although we know the irregularity results from a divergence between the topologies induced by the metric and the total ordering, there is no consensual quantitative measure which agrees with our visual perception. A quantitative measure can help to choose an appropriate ordering scheme for vector-valued mathematical morphology. In the following section, we propose to measure the irregularity using the Wasserstein metric.

The Wasserstein Metric and the Generalized Sum of Pixel-Wise Distances

The global irregularity index is defined in terms of the quotient of the Wasserstein metric and a generalized sum of pixel-wise distances. Let us begin by reviewing the generalized sum of pixel-wise distances.

Consider an image operator ψ : V → V. Given an input image I ∈ V, let J = ψ(I) denote the output of the image operator. The generalized sum of pixel-wise distances of I and J is an operator

D p : V × V → [0, +∞) given by D p (I, J) = x∈D d p I(x), J(x) 1 p , p ≥ 1.
(

) 5
The generalized sum of pixel-wise distances is one of the simplest measures that takes into account the metric d and the pixel locations. However, D p is usually not properly scaled; possibly because its dimension is the same as the metric d. For example, using the Euclidean RGB distance and p = 1, the images shown in Figure 1 yield the values D 1 (I, J L ) = 34.12 and D 1 (I, J M ) = 66.05. Note that D 1 (I, J L ) ≤ D 1 (I, J M ).

Hence, the generalized sum of pixel-wise distances is not an appropriate quantitative measure for the irregularity issue.

Let us now review the Wasserstein metric, also known as the earth mover's distance or the Kantorovich-Rubinstein distance in certain contexts [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF][START_REF] Villani | Topics in Optimal Transportation[END_REF]. The Wasserstein metric, named after the Russian mathematician Leonid Vaseršteȋn, has been previously used by Rubner et al. for content-based image retrieval [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF]. In the general case, the Wasserstein metric is used to compute distances between probability distributions. For discrete probabilities, however, it is formulated as a transportation problem.

The objective of a transportation problem is to minimize the cost to deliver items from n factories to m shops. In our context, the transportation problem minimizes the cost to transform the input image I into the output image J. The cost is defined using the metric on the value set V. Precisely, let V (I) = {v 1 , . . . , v n } and V (J) = {u 1 , . . . , u m } be the sets of color values of I and J, respectively. Given p ≥ 1, the cost to transform a value v i of I into a value u j of J is defined by

c ij = d p (v i , u j ), i = 1, . . . , n, j = 1, . . . , m. (6)
The Wasserstein metric, denoted by

W p : V × V → [0, ∞) for p ≥ 1, is given by W p (I, J) =   n i=1 m j=1 c ij x ij   1/p , p ≥ 1, (7)
where x ij solves the linear programming problem

                         minimize n i=1 m j=1 c ij x ij subject to m j=1 x ij = Card({x : I(x) = v i }), ∀i = 1, . . . , n, n i=1 x ij = Card({x : J(x) = u j }), ∀j = 1, . . . , m, x ij ≥ 0, ∀i = 1, . . . , n, ∀j = 1, . . . , m. (8)
In the transportation problem (8), the variable x ij represents the amount of the pixel value v i of I transformed to the pixel value u j of J. In some sense, the Wasserstein metric can be interpreted as the minimal cost to transform I into J. Considering p = 1 and the Euclidean distance, we obtain W 1 (I, J L ) = 6.18 and W (I, J M ) = 65.94 for the images shown in Figure 1. Note that the inequality W 1 (I, J L ) < W 1 (I, J M ) holds in this example. Like the generalized sum of pixel-wise distances, the Wasserstein metric is not an appropriate measure of the irregularity; possibly because it has the same dimension as the generalized sum of pixel-wise distances.

Global Irregularity Index

Although both generalized sum of pixel-wise distances and the Wasserstein metric are, per se, not appropriate to evaluate the irregularity issue, we advocate in this paper that their quotient yields a useful measure. First of all, note that the generalized sum of pixel-wise distances satisfies

D p (I, J) =   n i=1 m j=1 c ij y ij   1 p , p ≥ 1, (9)
where

y ij = Card ({x : I(x) = v i and J(x) = u j , x ∈ D}) , (10)
for all i = 1, . . . , n and j = 1, . . . , m. Moreover, it is not hard to see that

y ij ≥ 0, m j=1 y ij = Card({x : I(x) = v i }) and n i=1 y ij = Card({x : J(x) = u j }) (11)
for all i = 1, . . . , n and j = 1, . . . , m. Therefore, the generalized sum of pixelwise distances also measures the cost of transforming I into J. Moreover, D p and W p have the same units and magnitudes. Because W p is the minimal cost, the inequality W p (I, J) ≤ D p (I, J) holds for any I and J = ψ(I). Using these remarks, we propose to measure the irregularity using the mapping Φ g p : V × V → [0, 1] given by the relative gap between D p and W p . Precisely, given images I, J ∈ V, we define the global irregularity index by means of the equation

Φ g p (I, J) = D p (I, J) -W p (I, J) D p (I, J) , if D p (I, J) = 0, (12)
and Φ g p (I, J) = 0 if D p (I, J) = 0. Note that the larger the gap between W p (I, J) and D p (I, J), the larger the global irregularity index. Equivalently, we have

Φ g p (I, J) =    0, if D p (I, J) = 0, 1 - W p (I, J) D p (I, J) , otherwise. ( 13
)
The irregularity index is symmetric and bounded, that is, Φ p (I, J) = Φ p (J, I) and 0 ≤ Φ p (I, J) ≤ 1. Moreover, Φ g p (I, J) is a dimensionless quantity. The more irregular is J = ψ(I), the larger the value of Φ g p (I, J) is expect to be. For example, using p = 1 and the Euclidean distance, the irregularity index of the dilated images shown in Figure 1a) and 1c) are respectively Φ g 1 (I, J L ) = 81.90% and Φ g 1 (I, J M ) = 0.17%.

Average of Local Irregularity Indexes

Despite its mathematical formulation, computing the global irregularity index is not an easy task for natural images. Precisely, this irregularity index requires solving a linear programming problem with mn variables, where m and n are the number of distinct pixel values of the images I and J, respectively. In practical situations, the dimension of the linear programming problem ( 8) is extremely large, making it impossible to be solved in real time. To circumvent this computational burden, we propose to compute the Wasserstein metric and the generalized sum of pixel-wise distances locally and aggregate the values into a single quantitative index.

Let {W 1 , W 2 , . . . , W k }, with W ⊆ D for all = 1, . . . , k, be a family of possibly overlapping local windows such that D ⊆ ∪ k i=1 W i . Also, let D p (I , J|W ) and W p (I, J|W ) denote the generalized sum of pixel-wise distances and the Wasserstein metric computed restricting the images I and J to the local window W . The average of local irregularity indexes is defined by the following equation for all I, J ∈ V:

Φ a p (I, J) = 1 - k =1 W p (I, J|W ) D p (I, J|W ) 1 p (14)
We would like to emphasize that, because the irregularity index is given by a ratio, the geometric mean is used to aggregate the quotient of W p (I, J|W ) by D p (I, J|W ).

In our computational implementation, inspired by the structural similarity index (SSIM) [START_REF] Wang | Image quality assessment: From error visibility to structural similarity[END_REF], we used local square windows of size 8 × 8 with strides of 4 pixels. Using p = 1 and the Euclidean distance, the average of local irregularity indexes of the dilated images shown in Figure 1a) and 1c) are respectively Φ a 1 (I, J L ) = 15.27% and Φ a 1 (I, J M ) = 0.03%. Note that Φ a 1 (I, J L ) is significantly less than Φ g 1 (I, J L ) = 81.90%. However, the following experiment shows that the average of local irregularity indexes is highly correlated to the global irregularity index.

The Global Irregularity Index and the Average of Local Irregularity Indexes

Let us compare the global irregularity index and the average of local irregularity indexes using toy images similar to the one provided by Chevallier and Angulo [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF] but with different probabilities. Precisely, we first construct an image of size 64 × 32 with two stripes of the same width but with the colors black u = (0, 0, 0) and blue v = (0, 0, 1) in the RGB color space. Then, a pure black pixel value is replaced by the black w = (0.005, 0, 0) with probability π ∈ [0, 1]. The resulting image I is dilated by a cross structuring element using both the lexicographical and the marginal ordering schemes. The simulation has been repeated 100 times for each probability π ∈ [0, 1]. Figure 2 shows the mean of both the global irregularity index and the average of local irregularity indexes by the probability π. Again, we used the Euclidean distance to compute the cost to transform pixel values. For a better interpretation of this graph, dilated images obtained using the lexicographical ordering from images generated probabilities π = 0.0, 0.25, 0.50, 0.75, and 1.0 are shown at the bottom of Figure 2.

Note that both the global irregularity index and the average of the local indexes are very close to zero for the marginal ordering. In contrast, using the lexicographical RGB ordering, both irregularity indexes increase until close to π = 0.3 and then decrease. Furthermore, the irregularity indexes agree with the visual irregularity provided in the sample images at the bottom of Figure 2. Finally, we would like to point out that the correlation between the global and average of local irregularity indexes is 98.77%. Therefore, although they have different scales, we believe both the global irregularity index and the average of the local index can be used as an effective measure for the irregularity issue.

Example with Natural Color Images

Let us now provide some examples with natural color images. Precisely, we compute the average of local irregularity indexes Φ a p , the generalized sum of pixel-wise distances D p , and the average of local Wasserstein metrics, denoted by W a p , for several RGB images of size 256 × 256. Because of their computational burden, we refrained to compute the global irregularity index and the Wasserstein metric globally. In contrast to Φ a p , however, the local Wasserstein metrics are aggregated using the arithmetic mean. Moreover, for better scaling the D p and W a p , these quantities have been divided by the number of pixels of the processed image. Like in the previous examples, we used the Euclidean distance as the metric of the value set. The quantitative measures have been computed using the erosion, the dilation, the opening, and the closing by a 7×7 square structuring element. Two approaches based on the total ordering have been considered: One based on the RGB lexicographical order and the other based on a supervised reduced ordering [START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF]. In a supervised reduced ordering, the pixel values are ranked using a supervised machine learning technique trained on a set of background and foreground pixels. We also included the marginal approach for comparison purposes. Recall that the marginal approach is not based on a total order. Thus, it can circumvent the irregularity issue.

Figure 3 summarizes the outcome of this computational experiment. To facilitate the exposition, the average of the local irregularity indexes, the generalized sum of pixelwise distances divided by the number of pixels of the image, and the average of local Wasserstein metric divided by the number of pixels of the local windows, are presented as a triple (Φ a p , D p , W a p ) below the output images in percentage. The weakness of both the generalized sum of pixel-wise distances D p and the Wasserstein metric W p for measuring the irregularity is observed on the images shown for the marginal ordering. Similar remarks hold by comparing the irregularity indexes obtained for the images 3j), 3k), and 3l) or 3m), 3n), and 3o). In fact, except for the boat image, the irregularity index obtained for the lexicographical and supervised approaches are significantly larger than the irregularity index yielded by the marginal approach. We invite the reader to carefully compare the images and the corresponding quantities shown in Figure 3. We hope this example will help the reader to be convinced of the superiority of the proposed irregularity index, in particular, over both the generalized sum of pixel-wise distances and the Wasserstein metric.

Concluding Remarks

In this paper, we proposed two quantitative measures for the irregularity issue. Namely, the global irregularity index and the average of the local irregularity indexes, denoted respectively by Φ g p and Φ a p . Although Chevallier and Angulo provided a rigorous formulation of the irregularity issue [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF], as far as we know, there is no effective quantitative measure for this problem. For example, the generalized sum of pixel-wise distance D p , which is closely related to the norm of (internal or external) gradient or top-hat operations, is not a dimensionless measure [START_REF] Soille | Carving and adpative drainage enforcement of grid digital elevation models[END_REF][START_REF] Soille | Morphological Image Analysis[END_REF]. Similarly, the Wasserstein metric W p , which is eventually known as the earth mover's distance, is also not a dimensionless measure [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF]. Hence, they may not be appropriately scaled. Accordingly, as can be observed in Figure 3, both D p and W p are not appropriate for measuring the irregularity issue. In contrast, the dimensionless global irregularity index given by the relative gap between D p and W p yielded good quantitative values for the irregularity for images with few pixel values. In practical situations, however, the number of distinct pixel values makes it impossible to compute the global irregularity index. To circumvent this computational drawback, we proposed computing the geometric mean of the irregularity index on several small windows. Visual interpretations of the irregularity indexes are provided using both synthetic and natural images.

Finally, we would like to point out that the irregularity index can be used in the future to evaluate the performance of morphological operators. The irregularity index can also be used for the design of efficient morphological operators. For example, it can be used as the objective function for the design of vector-valued morphological operators based on uncertain reduced orderings [START_REF] Sangalli | Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings[END_REF].

Fig. 1 .

 1 Fig.1. Illustractive example of the irregularity issue. Image with three colors and its corresponding dilation by a cross structuring element using the RGB lexicographial and marginal orderings.

Fig. 3 .

 3 Fig. 3. Illustrative examples of the average of local irregularity indexes Φ a 1 computed for several color images using different morphological operators. For comparison purposes, we also included the sum of pixel-wise distances D1 and the average of local Wasserstein metric W a 1 .

Measuring the IrregularityIn this section, we present a quantitative measure for the irregularity issue, called the irregularity index. We begin by presenting a global irregularity index. Then, we propose computing the irregularity index as an average of local irregularity indexes. Although we are interested in measuring the irregularity implied by a total ordering, we will not assume V is totally ordered. Indeed, the proposed irregularity measure is well defined whenever D is finite and V is a metric space.

This work was supported in part by the São Paulo Research Foundation (FAPESP) under grant no 2019/02278-2 and the National Council for Scientific and Technological Development (CNPq) under grant no 310118/2017-4.

Dilated images

Fig. 2. Top: Irregularity indexes by the probability of replacing a pure black pixel value u = (0, 0, 0) by the black w = (0.005, 0, 0) in an image similar to Figure 1b). Bottom: Lexicographical dilation of images obtained using π = 0.0, 0.25, 0.5, 0.75, and 1.0, respectively. in Figure 3. As pointed out previously, D p and W p are not dimensionless measures; they are possibly measured using some photometric quantity such as lumen [START_REF] Trussell | Photometry and colorimetry[END_REF]. As a consequence, they may yield misleading values. For example, comparing the eroded boats and the dilated tulips depicted respectively in Figure 3b) and 3f), one may conjecture that the larger D p or W p , the more irregular is the image. However, opening the Lena image or closing the balloons images using the lexicographical ordering, yielded the irregular images shown in Figure 3j) and 3n) whose quantities D 1 and W a 1 are much smaller than the values obtained for the eroded boat images.

Let us now address the proposed irregularity index Φ a 1 , which is a dimensionless measure. Note that the three eroded versions of the boat image shown in Figure 3b), 3c), and 3d) are quite similar. Accordingly, the corresponding irregularity indexes have similar magnitudes. In contrast, the dilated version of the tulips image, depicted in 3f), 3g), and 3h), are quite different. In particular, the irregularities of the red pixels in the flowers on Figure 3f) obtained using the lexicographical ordering is noticeable. In agreement, the irregularity index Φ a 1 is much larger for the lexicographical ordering than