Alperen A Ergür
email: alperen.ergur@utsa.edu

Máté L Telek

Josué Tonelli-Cueto
email: josue.tonelli.cueto@bizkaia.eu

On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system f1(x) = 0, . . . , fn(x) = 0 of n random real polynomials in n variables, where each f k has a prescribed set of exponent vectors in a set A k ⊆ Z n of size t k . Assuming that the coefficients of the f k are independent Gaussian of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by 4 -n n k=1 t k (t k -1). This result is a probabilisitc version of Kushnirenko's conjecture; it provides a bound that only depends on the number of terms and is independent of their degree.

.

Introduction

Instances of polynomial system solving arise in a variety of problems coming from kinematics [START_REF] Sommese | The Numerical solution of systems of polynomials arising in engineering and science[END_REF], dynamical systems [START_REF] Fantuzzi | Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization[END_REF], mathematical modeling of chemical reaction networks [START_REF] Dickenstein | Algebraic geometry tools in systems biology[END_REF], computer vision [START_REF] Kukelova | Automatic generator of minimal problem solvers[END_REF], and computer aided geometric design [START_REF] Cox | Applications of Polynomial Systems[END_REF][START_REF] Sederberg | Algebraic geometry for computer-aided geometric design[END_REF]. The common thread among these plethora of applications is that the polynomial systems that occur are structured, and the solutions that are most useful are the real ones. In this setting, the basic question is: how many real zeros are there? When we consider zeros over complex numbers, we have the power of intersection theory [START_REF] Eisenbud | 3264 and all that-a second course in algebraic geometry[END_REF]. In particular, for sparse polynomial systems the number of complex solutions is given by the celebrated BKK bound [START_REF] Bernshtein | The number of roots of a system of equations[END_REF][START_REF] Kushnirenko | Polyèdres de Newton et nombres de Milnor [French[END_REF] (cf. [START_REF] Sottile | Real solutions to equations from geometry[END_REF]Chapter 3]). The story becomes more complicated and interesting for real zeros. Consider the following simple polynomial system:

a 1 + b 1 X + γ 1 Y + c 1 XY Z d = 0 a 2 + b 2 X + γ 2 Y + c 2 XY Z d = 0 a 3 + b 3 X + γ 3 Y + c 3 XY Z d = 0.
where d ∈ N and the a k , b k , c k and d k are real numbers. Although the system has, generically, d complex zeros, it has at most two real solutions. This example illustrates that the number of real zeros cannot be estimated using the number of complex solutions.

The above phenomenon is a general one. Khovanskiȋ, in his seminal book Fewnomials [44], showed that the maximum number of real zeros of a polynomial system can be bounded in terms of the description complexity of the system. The term fewnomial refers to a polynomial with few monomials, and so to a polynomial with a low description complexity. He showed that independent of the degree of its monomials, real fewnomial systems have few real zeros.

The work of Khovanskiȋ was an answer to several conjectures by Kushnirenko in 1977 [START_REF] Kushnirenko | Letter to Professor Sottile[END_REF]: A regular positive zero of a polynomial system f = (f 1 , f 2 , . . . , f n ) is a point x ∈ R n with positive coordinates (x i > 0) such that the Jacobian matrix of the polynomial system f at x, D x f , is full-rank. We denote the set of regular positive real zeros by Z r (f, R n + ). Among the conjectures of Kushnirenko, the following one remains widely open: 1 Kushnirenko's Question: Let A 1 , . . . , A n ⊂ N n be finite sets of sizes t 1 , . . . , t n and f the system of fewnomials given by

f 1 = α∈A1 f 1,α X α . . . f n = α∈An f n,α X α
Is there a bound of the form #Z r (f, R n + ) ≤ poly(t 1 , . . . , t n ) n ?

It is fair to say that Kushnirenko's question is one of the most challenging problems in real algebraic geometry. However, its reach and importance goes far beyond. In complexity theory, variants of Kushnirenko's question (where the bound should be explicit in an algebraic complexity measure), such as the real τ -conjecture [START_REF] Koiran | Shallow Circuits with High-Powered Inputs[END_REF] or the adelic τ -conjecture [START_REF] Phillipson | Fewnomial systems with many roots, and an adelic tau conjecture[END_REF], imply Valiant's algebraic variant of P vs NP (see [START_REF] Bürgisser | Completeness and reduction in algebraic complexity theory[END_REF] for details and relation to classical questions in complexity theory, and [START_REF] Briquel | The real tau-conjecture is true on average[END_REF] for a probabilistic take on the problem).

Specific cases of Kushnirenko's question play an important role in studying chemical reaction networks. In that setting, the steady states of the network correspond to positive real zeros of a certain parameterized polynomial system [START_REF] Dickenstein | Algebraic geometry tools in systems biology[END_REF]. Although there exist several methods to decide whether the system has at least two positive zeros [START_REF] Bihan | Lower bounds for positive roots and regions of multistationarity in chemical reaction networks[END_REF][START_REF] Müller | Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry[END_REF][START_REF] Perez Millan | Chemical Reaction Systems with Toric Steady States[END_REF][START_REF] Feliu | A computational method to preclude multistationarity in networks of interacting species[END_REF][START_REF] Feliu | Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations[END_REF], finding (tight) upper bounds is a hard and open problem. The polynomial system arising from a reaction network is usually sparse, and the maximum number of its positive zeros is much smaller compared to the number of complex zeros [START_REF] Gross | Algebraic systems biology: a case study for the Wnt pathway[END_REF][START_REF] Obatake | Oscillations and bistability in a model of ERK regulation[END_REF]. Thus techniques from real algebraic geometry are required to have insight into the number of steady states [START_REF] Feliu | Multistationarity and bistability for Fewnomial chemical reaction networks[END_REF][START_REF] Giaroli | Parameter regions that give rise to 2⌊ n 2 ⌋ + 1 positive steady states in the n-site phosphorylation system[END_REF][START_REF] Bihan | Lower bounds for positive roots and regions of multistationarity in chemical reaction networks[END_REF].

Khovanskiȋ [44, §3.14, Corollary 4] obtained a bound of the form

2 ( t-1 2 ) (n + 1) t-1
where t = # (∪ n k=1 A k ) is the number of types of monomials appearing in the system. Later, Bihan and Sottile [START_REF] Bihan | New fewnomial upper bounds from Gale dual polynomial systems[END_REF][START_REF] Bihan | Fewnomial bounds for completely mixed polynomial systems[END_REF] would improve this bound to

e 2 + 3 4 2 ( t-n-1 2 ) n t-n-1
and, in a special mixed case in which t = n k=1 t k -n + 1, further improved this bound to

e 2 + 3 4 2 ( t-n-1 2 ) t -n -1 t 1 -2, . . . , t n -2 .
After four decades of hard work, as of today, we are still far from answering Kushnirenko's question. We only have precise bounds for special cases [START_REF] Bihan | A sharp bound on the number of real intersection points of a sparse plane curve with a line[END_REF][START_REF] Koiran | On the intersection of a sparse curve and a low-degree curve: a polynomial version of the lost theorem[END_REF][START_REF] Li | Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces[END_REF], and even the bivariate case of Kushnirenko's question remains open [START_REF] Koiran | On the intersection of a sparse curve and a low-degree curve: a polynomial version of the lost theorem[END_REF]. We refer the reader to [START_REF] Sottile | Real solutions to equations from geometry[END_REF] for a comprehensive exposition, to [START_REF] Dickenstein | Positive solutions of sparse polynomial systems[END_REF] for a short survey and to [74, Appendix F, §1] for a historical survey.

Our current inability to answer Kushnirenko's question, and the pressing need for an answer in applications, motivates a probabilistic take on the problem. How many zeros of a random real fewnomial system are real? This question turns out the be harder than it sounds. There is a welldeveloped theory (either relying on the kinematic formula [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF][START_REF] Edelman | How many zeros of a random polynomial are real?[END_REF] or convex bodies [START_REF] Breiding | The zonoid algebra, generalized mixed volumes, and random determinants[END_REF][START_REF] Ya | Average Number of Roots of Systems of Equations[END_REF][START_REF] Mathis | Expectation of a random submanifold: the zonoid section[END_REF]) in random real algebraic geometry. This theory has produced strong results on the expected number of real zeros of random dense polynomials, that is, random polynomials supported on all monomials of a given degree [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF][START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF][START_REF] Armentano | Asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Letendre | Variance of the volume of random real algebraic submanifolds[END_REF][START_REF] Letendre | Variance of the volume of random real algebraic submanifolds II[END_REF][START_REF] Armentano | Central limit theorem for the number of real roots of Kostlan Shub Smale random polynomial systems[END_REF][START_REF] Armentano | Central limit theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems[END_REF]. However, in the case of random structured polynomials, applying the theory becomes more complicated [START_REF] Breiding | Average degree of the essential variety[END_REF][START_REF] Ya | How many roots of a system of random Laurent polynomials are real?[END_REF], and earlier available results are either for random polynomials supported on a product of simplices [START_REF] Rojas | On the average number of real roots of certain random sparse polynomial systems[END_REF] or depend strongly on the degree of the monomials [START_REF] Malajovich | Polynomial systems and the momentum map[END_REF] 1 . In summary, the aforementioned results in random real algebraic geometry do not address Kushnirenko's question. In [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF], we obtained a bound for random fewnomial systems with fixed support (A = A 1 = • • • = A n ) and centered Gaussian random coefficients with the same variance structure. In our view, the importance of the bound in [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF] relies on two facts:

(1) The bound depends only in the number of distinct monomials.

(2) The bound does not depend on the variance structure of the random coefficients. This latter point should be stressed, since, for random dense polynomial systems all available results place strong assumptions on the variance structure.

Recently, Bürgisser [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF] generalized the results of [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF] to the mixed case (where A 1 , A 2 , . . . , A n are not necessarily equal). However, the result in [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF] only holds for centered Gaussian fewnomial systems in which all variances are equal to one. In this paper, we obtain estimates that hold for arbitrary support sets and arbitrary variances in the same spirit with Kusnirenko's question.

The simplest form of our result can be seen in the theorem below. In the remainder of the introduction we explain the consequences of our result in specific cases and state our main theorem in full technical detail. We also provide side results on the volume of random fewnomial varieties.

Theorem 1.1. Let A 1 , . . . , A n ⊂ R n be finite sets of sizes t 1 , . . . , t n , and f the system of n random fewnomials given by

f 1 = α∈A1 f 1,α X α . . . f n = α∈An f n,α X α (1.1)
where the f k,α are independent centered Gaussian random variables (with no restrictions placed on variances). Then we have

E f #Z(f, R n + ) ≤ 1 4 n n k=1 t k (t k -1) (1.2)
where

Z(f, R n + ) is the zero set of f in R n + . Remark 1.2.
Observe that if the affine span of n k=1 A k is not the whole R n (or, more generally, if A 1 , . . . , A n does not contain an independent transversal (see [76, Lemma 1] for the precise definition)), then Z(f, R n + ) = ∅ with probability one.

Random Mixed Fewnomials with Restricted Variances

The following theorem shows that the bound of Theorem 1.1 can be improved when we impose restrictions on the variances on the random fewnomial system. This result improves the main theorem of Bürgisser [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF]Theorem 1.1] by allowing more flexible assumptions on the variance structure and a better multiplying factor: 4 -n instead of (2π) -n/2 . We give the proof in Section 3.

Theorem 1.3. Under the same notations and assumptions of Theorem 1.1, assume that the variance vectors

v k = (v k,α ) α∈A k of the f k satisfy the following conditions: (VM1) for all k and all α ∈ A k , v k,α ≤ 1.
(VM2) for all k and α ∈ A k a vertex of the polytope

P k := conv(A k ), v k,α = 1. Then E f #Z r (f, R n + ) ≤ 1 4 n V n k=1 P k n k=1 (t k -1) (1.3)
where V ( n k=1 P k ) is the number of vertices of the Minkowski sum n k=1 P k . Remark 1.4. Interestingly, in the univariate case (n = 1), the obtained upper bound, (t 1 -1)/2, agrees with the average value of the bound given by Descartes' rule of signs [START_REF] Descartes | La Géométrie. Digital reproduction of 2008 by Project Gutenberg[END_REF]. However, optimal results in the univariate case, when all variances are equal to one, were already obtained in [START_REF] Jindal | How many zeros of a random sparse polynomial are real?[END_REF].

Random Unmixed Fewnomials with Restricted Variances

In the case where the system is unmixed and some restrictions are imposed on the variances we obtain better bounds. This result improves the main theorem of [22, Theorem 1.2] (see Remark 1.6 for details). We give the proof of this in Section 4.

Theorem 1.5. Under the same notations and assumptions of Theorem 1.1, assume that

A = A 1 = • • • = A n
for some A ⊂ R n of size t, and assume that the variance vectors v k of the f k satisfy either the assumptions (VM1) and (VM2) in Theorem 1.3 or

v 1 = • • • = v n .
(1.4) A specific case that attracted considerable attention is when #A = n + ℓ and ℓ is a fixed constant [START_REF] Bihan | New fewnomial upper bounds from Gale dual polynomial systems[END_REF], specially when ℓ = 2 [START_REF] Bihan | Polynomial systems supported on circuits and dessins d'enfants[END_REF][START_REF] Bihan | Descartes' rule of signs for polynomial systems supported on circuits[END_REF][START_REF] Rojas | Counting Real Roots in Polynomial-Time via Diophantine Approximation[END_REF]. The following corollary (whose proof is at the end of Section 4) shows that if such systems are generated randomly then it is hard to find any positive zero at all. Corollary 1.7. Under the same the same notations and assumption of Theorem 1.5, assume that for some constant ℓ ∈ N,

Then E f #Z(f, R n + ) ≤ n + 1 4 n t n + 1 . ( 1
#A = n + ℓ Then P f Z(f, R n + ) = ∅ ≤ ℓ(n + 1) ℓ 4 n . (1.6)
Moreover, since ℓ is a fixed constant, we have that

lim n→∞ P f Z(f, R n + ) = ∅ = 0.
Remark 1.8. If we substitute the constant ℓ by logpolynomial function ℓ(n) = O(log a (n)) with a > 0 a constant, the limit probability would still be zero.

Main Result in Full Technical Detail

Recall that given a finite set A ⊂ R n and a map

l : A → R,
the upper envelope of A with respect l is the convex polyhedron given by

L(A, l) := conv l(α) -s α | α ∈ A, s ≥ 0 . (1.7)
Intuitively, this set arises when we lift A according to l and we look at the convex hull of these lifted points,

P l := conv l(α) α | α ∈ A ,
from above. We can now state our main theorem, from which Theorems 1.1 and 1.3 will also follow.

Theorem 1.9. Let A 1 , . . . , A n ⊂ R n be finite sets of sizes t 1 , . . . , t n , and f the the system of n random fewnomials given by

f 1 = α∈A1 f 1,α X α . . . f n = α∈An f n,α X α
where the f k,α are independent centered Gaussian random variables. Consider, for each f k , its variance vector

v k = (v k,α ) α∈A k
and construct the lifting functions l k : A k → R given by

l k : α → 1 2 ln v k,α . (1.8)
Then we have

E f #Z r (f, R n + ) ≤ 1 4 n V n k=1 L(A k , l k ) n k=1 (t k -1) (1.9)
where V (

n k=1 L(A k , l k )) is the number of vertices of the Minkowski sum n k=1 L(A k , l k ). We note that one can interpret the expression V n k=1 L(A k , l k )
as the number of elements of n k=1 A k that are used in the regular mixed subdivision induced by the lifting functions l 1 , . . . , l n which are defined by the logarithm of the variances. Indeed, we find the theory of regular (mixed) subdivisions [START_REF] De Loera | Triangulations: Structures for algorithms and applications[END_REF] at the core of our development. Subdivisions have been used earlier in the work of Bihan, Santos, and Spaenlehauer [START_REF] Bihan | A polyhedral method for sparse systems with many positive solutions[END_REF] and Sturmfels [START_REF] Sturmfels | On the number of real roots of a sparse polynomial system[END_REF] for constructing fewnomial systems with many zeros, in the work of Sturmfels on the Newton polytope of A-resultants [72, Sect. 2], in the seminal work by Gelfand, Kapranov and Zelevinsky [START_REF] Gelfand | Discriminants, resultants and multidimensional determinants[END_REF], and implicitly in Viro's patchworking method [START_REF] Viro | From the sixteenth Hilbert problem to tropical geometry[END_REF] (see [START_REF] Ergür | A polyhedral homotopy algorithm for real zeros[END_REF]Section 2.2] for an exposition). We don't have any good explanation for why mixed regular subdivisions appear in random fewnomial theory, but there seems to be a deep connection to be uncovered.

Bounds on Volume of Random Projective Fewnomial Varieties

In complex algebraic geometry, the (normalized) volume of a complex algebraic variety is just the degree [START_REF] Bürgisser | Condition: The Geometry of Numerical Algorithms[END_REF]Corollary 20.10]. In real algebraic geometry, we can only talk about the average volume of a random variety-(normalized) volume is also referred to as "average degree" [START_REF] Bürgisser | Probabilistic Schubert calculus[END_REF]. Shub and Smale [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF] showed that for a random KSS homogeneous polynomial system f of degrees d 1 , . . . , d q in the n + 1 variables X 0 , X 1 , . . . , X n , we have that

E f vol n-q Z(f, P n R ) = vol n-q P n-q R vol n P n R q k=1 d k , (1.10)
where vol k P k R is the volume under the usual Riemannian structure. Moreover, we even have a central limit theorem in the case that all degrees are equal [START_REF] Armentano | Central limit theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems[END_REF]. The following result shows that, for random projective varieties given by random fewnomials, the average volume can be bounded independently of the degree. We give the proof in Section 5.

Theorem 1.10. Let q ≤ n, d 1 , . . . , d q ∈ N and A 1 , . . . , A q ⊂ N n+1 be finite subsets of sizes t 1 , . . . , t q such that for all k, {d k e 0 , . . . ,

d k e n } ⊆ A k ⊆ d k ∆ n
where {e 0 , . . . , e n } is the standard basis of R n+1 and ∆ n := conv(e 0 , . . . , e n ) the n-simplex. Consider the random homogeneous fewnomial system f given by

f 1 = α∈A1 f 1,α X α . . . f q = α∈Aq f q,α X α
where the f k,α are independent centered Gaussian random variables (with no restrictions placed on the variances). Then, with probability one, Z(f, P n R ) is of dimension at most n -q and

E f vol n-q Z(f, P n R ) ≤ vol n-q P n-q R vol n P n R (n(n + 1)) n-q 2 n q k=1 t k (t k -1) (1.11)
Remark 1.11. In Theorem 1.10, we are imposing conv(A k ) = d k ∆ n to guarantee that Z(f, P n R ) does not contain a coordinate hyperplane with probability one. Although this can be achieved under more general conditions, we prioritize a simple statement.

Remark 1.12. By our assumption, we have t k ≥ n + 1. Hence, when degree is large, for example, when

t k ≤ 2 n/q (n(n + 1)) n/q-1 d k ,
(1.11) gives a better bound than (1.10). In particular, the random projective fewnomial hypersurface given by n i=0 f i X d i has an average volume bounded by

vol n-q P n-q R vol n P n R n(n + 1) 2 n ,
which does not depend on the degree.

Overview and Outline of the Proof

Our proof follows a similar path to the proof in [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF]. We first prove an integral-probabilistic formula for the number of zeros which depends on the determinant of a random matrix. To bound this complicated integral, we use the Cauchy-Binet formula2 and express the determinant as a summation of many determinants of n × n matrices. In turn, this splits the integral formula into a sum of integrals over determinants of n × n matrices. We bound these summands by interpreting them as expected number of real zeros of a specific system of random polynomials and bounding these expectations simply by the maximum number of zeros of these special systems.

Expanding the above, we prove the integral-probabilistic formula (2.2) in Section 2 using kinematic formulas [START_REF] Bürgisser | Probabilistic Schubert calculus[END_REF][START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF] and a measure theoretic result of Appendix A. After this, in Section 3, we use the Cauchy-Binet formula [START_REF] Broida | A comprehensive introduction to linear algebra[END_REF]Theorem 4.15] to split the integral into several summands. However, the proof complicates and we need to do a polyhedral subdivision-induced by the normal fan of n k=1 L(A k , l k ) in the statement of Theorem 1.9-to perform the reduction of the random general case to the deterministic case-a binomial system-analyzed in Proposition 3.2.

Although the above reduction from the general random case to special deterministic cases plays a central role in our proof, a direct analytical proof is possible. We provide a fully analytical proof of the Proposition 3.2 to show this fact.

Organization

In Section 2 we obtain a Rice formula for the expected number of real zeros. Section 3 proves our main result. Section 4 deals with unmixed systems, and Section 5 contains the proof of volume estimate for random fewnomial varieties.

A Rice Formula for the Expected Number of Zeros

At the center of [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF], we find an integral-probabilistic formula for the number of expected zeros of a random system. This formula is only implicitly stated in a special case (the case where ψ is injective). We make this formula explicit and prove it in its full generality. We do this by combining the kinematic formulas [START_REF] Bürgisser | Probabilistic Schubert calculus[END_REF][START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF] used in [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF] with standard arguments of measure theory, such as Dynkin's lemma [1, p. 4.11]. We note that the formula below is a special case of Rice formula [5, Theorem 6.2]. We provide a proof using the kinematic formula for completeness and because this proof is interesting in its own right.

Recall that for f :

U ⊆ R n → R n with U ⊆ R n open, Z r (f, U ) := {ζ ∈ U | f (x) = 0, det D x f (x) = 0}
is the set of regular zeros of f in U . The following theorem gives the promised integral-probabilistic formula for the number of zeros of a random system where each equation is generated independently.

Theorem 2.1 (Rice formula). Let Ω ⊆ R n be an open set and let

ϕ k : Ω → R m k +1
be smooth maps such that the map ψ : Ω → n k=1 S m k given by

ψ : x →    ψ 1 (x) . . . ψ n (x)    :=    ϕ 1 (x)/ ϕ 1 (x) . . . ϕ n (x)/ ϕ n (x)    (2.1)
is well-defined. Consider a random system of equations f 1 (x) = 0, . . . , f n (x) = 0 given by

f k := m k +1 i=0 c k,i ϕ k,i
where the c k,i are i.i.d. standard Gaussian random variables. Then, for every Borel measurable set Ω ⊆ Ω,

E c #Z r (f, Ω) = (2π) -n/2 Ω E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    dx (2.2)
where the a k are independent standard Gaussian vectors in ψ k (x) ⊥ ⊂ R m k +1 , i.e., the subspace orthogonal to ψ k (x).

Corollary 2.2. Under the same assumptions and notations of Theorem 2.1, for every Borel measurable set Ω ⊆ Ω,

E c #Z r (f, Ω) = (2π) -n/2 Ω 1 n k=1 ϕ k (x) E a det    a T 1 D x ϕ 1 . . . a T n D x ϕ n    dx (2.3)
where the a k are independent standard Gaussian vectors in the subspace ϕ k (x) ⊥ .

Corollary 2.3. Under the same assumptions and notations of Theorem 2.1, if

m 1 = m 2 = • • • = m n = 1
and, for some ξ : Ω → R n ,

ϕ 1 = 1 ξ 1 (x) , . . . , ϕ n = 1 ξ n (x)
then, for every Borel measurable set Ω ⊆ Ω,

E c #Z r (f, Ω) = 1 π n Ω |det D x ξ| n k=1 (1 + ξ k (x) 2 ) dx (2.4)
Proof of Theorem 2.1. We will show later that without loss of generality we can assume that ψ : Ω → n k=1 S m k is an embedding, i.e., it satisfies: (I1) ψ is an immersion, i.e., for all x ∈ Ω, rank D x ψ = n.

H k := x ∈ R m k +1 | m k i=0 c k,i x k,i = 0
which are uniformly distributed in the corresponding Grassmannian (because the c k,i are i.i.d. standard Gaussians). Observe that

#Z(f, Ω) = #ψ(Ω) ∩ (H 1 × • • • × H n ) , (2.5) since ψ(x) ∈ H 1 × • • • × H n if
ψ(Ω) ∩ (H 1 × • • • × H n )
is transversal with probability one, and hence it is zero-dimensional almost surely. Hence, with probability one,

#Z r (f, Ω) = #Z(f, Ω) = #ψ(Ω) ∩ (H 1 × • • • × H n ) ,
and so

E c #Z r (f, Ω) = E H1,...,Hn #ψ(Ω) ∩ (H 1 × • • • × H n ) .
Now, by the kinematic formula stated in [20, Theorem 3.2] (cf. [START_REF] Bürgisser | Probabilistic Schubert calculus[END_REF][START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF]) and [START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF]Lemma 3.4.], the equation (2.2) holds for Ω = Ω.

We now show that we can assume (I1), (I2) and (O) without loss of generality. If (I1) does not hold, then the set

V := {x ∈ Ω | rank D x ψ < n}
is non-empty. Now for a given x ∈ V , note that

D x ψ k = 1 ϕ k (x) (I -ψ k (x)ψ k (x) T )D x ϕ k . (2.6)
So, to have D x ψ < n we have to have a v x ∈ R n \ 0 in the kernel of D x ψ for which we have that for all k

D x ϕ k (v x ) ∈ Rϕ k (x).
That is there exist v x ∈ R n \ 0 and t 1 , . . . , t n ∈ R such that for all k,

D x ϕ k (v x ) = t k ϕ k (x).
Translating this into D x f we conclude that there are t k ∈ R that satisfies

D x f(v x ) =    c T 1 D x ϕ 1 (v x ) . . . c T n D x ϕ n (v x )    =    t 1 c T 1 ϕ 1 (x) . . . t n c T n ϕ n (x)    =    t 1 f 1 (x) . . . t n f n (x)    .
This means f cannot have regular zeros at x for any x ∈ V . Hence,

#Z r (f, V ) = 0 and #Z r (f, Ω) = #Z r (f, Ω \ V ).
Moreover, since for all x ∈ V , rank D x ψ < n, we have that for all

a 1 ∈ R m1+1 , . . . , a n ∈ R mn+1 , det    a T 1 D x ψ 1 . . . a T n D x ψ n    = 0
and so

Ω E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    dx = Ω\V E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    dx.
Hence, taking Ω \ V instead of Ω, we can assume, without loss of generality, that (I1) holds. Assume now that (I1) holds. We will show how we can assume (I2) and (O) without loss of generality. We will make use of Theorem A.4, which relies on standard arguments in measure theory and which we prove in Appendix A. By (I1), the map ψ : Ω → S n is an immersion and so, by [START_REF] Lee | Introduction to smooth manifolds[END_REF]Proposition 5.22], it is a local embedding, i.e., for each x ∈ Ω, there is

r x > 0 such that ψ : B(x, r x ) ⊆ Ω → n k=1 S m k is injective. Therefore the collection of open sets U := {U ⊆ Ω | U open, U ⊆ Ω, U compact and ψ |U injective}.
(2.7)

satisfies that for all x ∈ Ω, there is r x > 0 such that for all r < r x , B(x, r) ∈ U. Hence U is a base for the topology of Ω-assumption (U0) of Theorem A.4. Moreover, U is closed under containment of open sets-assumption (U1) of Theorem A.4. Consider the following two Borel measures:

µ : B → E c #Z r (f, B)
and

ν : B → (2π) -n/2 B E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    dx.
We only need to show that µ = ν. Once this is done, (2.2) holds in the desired generality.

For each U ∈ U, (I2) holds. Thus we have that (2.2) holds with Ω = U , and so

µ(U ) = ν(U )
-assumption (U4) of Theorem A.4. Moreover, this quantity is finite-assumption (U3) of Theorem A.4. To see this, note that

E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    ≤ E a n k=1 a k 2 E a det    âT 1 D x ψ 1 . . . âT n D x ψ n    2
where âk := a k / a k is uniformly distributed in S m k ∩ ψ k (x) ⊥ ., and so, by compactness, for x ∈ U ,

E a det    a T 1 D x ψ 1 . . . a T n D x ψ n    ≤ E a n k=1 a k 2 max âk ∈S m k x∈U det    âT 1 D x ψ 1 . . . âT n D x ψ n    < ∞.
Thus the integral of the left-hand side over U ⊂ U , and so ν(U ), must be finite. Now, Ω is σ-compact, because it is an open subset of R n and all hypothesis of Theorem A.4 ((U0), (U1), (U2) and (U3)) are satisfied, so µ = ν as we wanted to show.

Remark 2.4. The above proof can be applied to extend [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF] to functions that are not semialgebraic and this immediately extends the result in [START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF] to arbitrary real exponents.

Proof of Corollary 2.2. This follows from Theorem 2.1 using (2.6), the properties of the determinant and that a k (I -

ψ k (x)ψ k (x) T ) = a k due to a k ∈ ψ k (x) ⊥ = ϕ k (x) ⊥ .
Proof of Corollary 2.3. We just have to apply Corollary 2.2. Fix x ∈ Ω, we only have to show that

(2π) -n 2 E a det    a T 1 D x ϕ 1 . . . a T n D x ϕ n    = 1 π n | det D x ξ| n k=1 1 + ξ k (x) 2 )
where the a k ∈ ψ k (x) ⊥ are independent standard Gaussian random vectors. Since we are in R 2 , we have that

ϕ k (x) ⊥ = span 1 √ 1+ξ k (x) -ξ k (x) 1
and so we can write

a k = z k 1 + ξ k (x) -ξ k (x) 1 with the z k ∈ R i.i.d. standard Gaussians. Moreover, because of this, a k D x ϕ k = z k √ 1+ξ k (x) 2 D x ξ k , and so det    a T 1 D x ψ 1 . . . a T n D x ψ n    = n k=1 |z k | | det D x ξ| n k=1 (1 + ξ k (x) 2 )
. Now, an easy computation shows that E z n k=1 |z k | = (2/π) n/2 , and the claim follows.

We finish this section with the following two propositions which will allow us to drop the regularity assumption from our theorems. Proposition 2.5. Under the same assumptions and notations of Theorem 2.1, if for all x ∈ Ω, rank D x ψ = n, then, for every Borel measurable set Ω ⊆ Ω,

Z(f, Ω) = Z r (f, Ω)
with probability one. Proposition 2.6. Under the same analogous notations of Theorem 2.1, assume that the map the map ψ : Ω → q k=1 S m k , with q > n, given by

ψ : x →    ψ 1 (x)
. . .

ψ n (x)    :=    ϕ 1 (x)/ ϕ 1 (x) . . . ϕ q (x)/ ϕ q (x)    (2.8)
is well-defined and that for all x ∈ Ω, rank D x ψ = n.

Consider the random overdetermined system f 1 (x) = 0, . . . , f q (x) = 0 given by

f k := m k +1 i=0 c k,i ϕ k,i
where the c k,i are i.i.d. standard Gaussian random variables. Then

Z(f, Ω) = ∅
with probability one.

Proof of Proposition 2.5. The proof is as the proof of Theorem 2.1, but now we don't need to remove any subset from Ω so that that (I1) holds. Hence, arguing as before using (2.5) with the cover U, we have, by Sard's Theorem [59, §2] (cf. [START_REF] Bürgisser | Condition: The Geometry of Numerical Algorithms[END_REF]Proposition A.18]), that for all U ∈ U,

Z(f, U ) = Z r (f, U )
with probability one. Since this holds for all open sets in an open cover of Ω, it holds for Ω too.

Proof of Proposition 2.6. The proof is analogous to that of Proposition 2.5. However, when writing down the analogous of (2.5), we have that we are intersecting q > n hyperplanes with an ndimensional submanifold in q k=1 S m k . By Sard's Theorem [59, §2] (cf. [START_REF] Bürgisser | Condition: The Geometry of Numerical Algorithms[END_REF]Proposition A.18]), this intersection is almost surely transversal which implies that this intersection is empty as desired.

Proof of the Main Theorem

We now prove Theorem 1.9 which yields Theorems 1.1 and 1.3 as direct corollaries. We also prove Proposition 3.1, which allows us to remove the regularity from our statements for random fewnomial systems; and the key Proposition 3.2 that plays a central role in the proof of the main theorem. Proposition 3.1. Under the notations and assumptions of Theorem 1.9,

Z(f, R n + ) = Z r (f, R n + )
with probability one.

Proposition 3.2. Let γ 1 , . . . , γ n ∈ R n , s 1 , . . . , s n ∈ R and consider the following the random binomial system h given by the equations

h 1 = a 1 exp(γ t 1 X + s 1 ) + b 1 . . . h n = a n exp(γ t n X + s n ) + b n (3.1)
where the a k and b k are i.i.d. standard Gaussian random variables, and the following Borelmeasurable set

B := {x ∈ R n | for all k, γ T k x + s k ≤ 0}. Then 1 π n B n k=1 exp(γ T k x + s k ) n k=1 (1 + exp(2γ T k x + 2s k )) |det(γ 1 • • • γ n )| dx = E h #Z r (h, B) ≤ 1 4 n .
Remark 3.3. If the γ k are linearly independent, then the inequality in Proposition 3.2 is an equality.

Proof of Theorem 1.9. By Proposition 3.1, we can just bound the number of expected regular zeros. Instead of a system of fewnomials, we consider the equivalent system of exponential sums g = (g 1 , . . . , g n ) defined as

g 1 = α∈A1 c 1,α exp(α T X + l 1 (α))
. . .

g n = α∈An c n,α exp(α T X + l n (α)) (3.2)
where the c k,α are i.d.d. standard Gaussian variables. Observe that

E c #Z r (g, R n ) = E f #Z r (f, R n + ).
So we will need to bound the expected number zeros of g on R n . We do this by decomposing R n into a collection of polyhedral cells, and bounding the number the number zeros on each of these polyhedral cells.

To define the polyhedral cell, for α 1 ∈ A 1 , . . . , α n ∈ A n , consider

M (α 1 , . . . , α n ) := {x ∈ R n | for all k and for all α ∈ A k , α T x + l k (α) ≤ α T k x + l k (α k )}. (3.3)
Observe that z ∈ M (α 1 , . . . , α n ) if and only if the map

s x → 1 z T s x = s + z t x
attains, for each k, its maximum value inside L(A k , l k ) at

l k (α k ) α k , i.e., if 1 z is, for each k, in the normal cone of L k (A k , l k ) at l k (α k ) α k .
This shows that M (α 1 , . . . , α n ) is a full-dimensional polyhedral cell if and only if

n k=1 l k (α k ) α k is a vertex of the Minkowski sum n k=1 L(A k , l k ).
Moreover, since M (α 1 , . . . , α n ) is obtained by maximizing the value of linear functional over a finite collection of points, we have

R n = {M (α 1 , . . . , α n ) | α 1 ∈ A 1 , . . . , α n ∈ A n }. (3.4) 4 n n k=1 (t k -1). (3.5)
Observe that the solutions of g in any region do not change if we multiply each equation by exp -α T k xl k (α k ) . Thus, without loss of generality, we can assume that α 1 = • • • = α n = 0 and that l k (α k ) = 0. Thus by considering M := M (0, . . . , 0), ϕ k (x) = exp(α T x + l i (α)) α∈A k , and applying Corollary 2.2, we get

E c #Z r (g, M ) = (2π) -n 2 M 1 n k=1 ϕ k (x) E a det    a T 1 D x ϕ 1 . . . a T n D x ϕ n    dx (3.6)
where the a k ∈ ϕ k (x) ⊥ are independent standard Gaussian random vectors. Observe that D x ϕ k,0 = 0 since ϕ k,0 = 1. Thus, by the Cauchy-Binet formula [START_REF] Broida | A comprehensive introduction to linear algebra[END_REF]Theorem 4.15] applied to

   a T 1 D x ϕ 1 . . . a T n D x ϕ n    =    a T 1 . . . a T n    D x ϕ, we obtain det    a T 1 D x ϕ 1 . . . a T n D x ϕ n    = β1∈A1\{0},...,βn∈An\{0} n k=1 a k,β k det    D x ϕ 1,β1 . . . D x ϕ n,βn    ,
since, to get a full-rank minor of diag(a T k ), we need to pick a column corresponding to each a T k . So, by the triangle inequality we have det

   a T 1 D x ϕ 1 . . . a T n D x ϕ n    ≤ β1∈A1\{0},...,βn∈An\{0} n k=1 |a k,β k | det    D x ϕ 1,β1 . . . D x ϕ n,βn    . (3.7)
Now we use following observation: The coordinate projection

R A k → R x → x β k induces a linear functional λ k : ϕ k (x) ⊥ → R which has the norm λ k = 1 - ϕ k,β k (x) 2 ϕ k (x) 2 = ϕ k (x) 2 -ϕ k,β k (x) 2 ϕ k (x) . Thus a k,β k = λ β k (a k
) is a centered Gaussian random variable with variance λ k 2 , because it is the result of projecting the standard Gaussian random vector a k ∈ ϕ k (x) ⊥ with the linear functional λ k of norm λ k . Therefore

E a |a k,β k | = ϕ k (x) 2 -ϕ k,β k (x) 2 ϕ k (x) 2 π . (3.8)
Substituting (3.8) and D x ϕ k,β k = exp(β T k x + l k (β k ))β T k back in (3.6) combined with (3.7), we get E c #Z r (g, M ) ≤ β k ∈A k \{0} 1 π n M n k=1 ϕ k (x) 2 -ϕ k,β k (x) 2 exp(β T k x + l k (β k )) ϕ k (x) 2 | det(β 1 • • • β n )| dx.
If we bound each of the summands in the right-hand side by 1/4 n , we are done, since there are at most n k=1 (t k -1) many summands. We will use Proposition 3.2 with γ k = β k and s k = l k (β k ) to bound each summand by 1/4 n . To do this, we only need to show that

n k=1 ϕ k (x) 2 -ϕ k,β k (x) 2 ϕ k (x) 2 ≤ 1 n k=1 (1 + exp(2β T k x + 2l k (β k ))) . (3.9)
So, it suffices to show that for each k we have

ϕ k (x) 2 -ϕ k,β k (x) 2 ϕ k (x) 2 ≤ 1 (1 + exp(2β T k x + 2l k (β k ))) . (3.10)
We define

u k (x) := ϕ k,β k (x) = exp(β T k x + l k (β k )) and v k (x) = ϕ k (x) 2 -ϕ k,β k (x) 2 = α∈A k \{0,β k } exp(2α T x + 2l k (α)).
We only need to show then that

1 + v k (x) 2 1 + u k (x) 2 + v k (x) 2 ≤ 1 1 + u k (x) 2 , which is equivalent to 1 + v k (x) 2 2 -(1 + u k (x) 2 ) 1 + v k (x) 2 + u k (x) 2 ≥ 0. Observe that the polynomial T 2 -(1 + u k (x) 2 )T + u k (x) 2 has two roots: 1 and u k (x) 2 . Since 1 + v k (x) 2 ≥ 1, the above inequality is valid independently of v k (x) as long as u k (x) ≤ 1, which is precisely what happens for x ∈ M since M = M (0, . . . , 0) = {x ∈ R n | for all k, for any β k ∈ A k , β T k x + l k (β k ) ≤ 0}.
Hence, (3.9) holds in M , and by Proposition 3.2,

1 π n M n k=1 ϕ k (x) 2 -ϕ k,β k (x) 2 exp(β T k x + l k (β k )) ϕ k (x) 2 | det(β 1 • • • β n )| dx ≤ 1 4 n .
which completes the proof.

Proof of Proposition 3.1. Assume that the affine span of n k=1 A k is the whole of R n . By Propositions 2.5, we only need to show that for the map

ψ = (ϕ 1 / ϕ 1 , . . . , ϕ n / ϕ n ) where ϕ k (x) = (exp(α T x + l k (α))) α∈A k ,
satisfies that for all x ∈ R n , rank D x ψ = n. Note that working with this exponential formulation is enough as it does not affect the regularity of the zeros of a fewnomial system. Now, by (2.6), we have that rank D x ψ < n if and only if there is v x ∈ R n \ {0} such that for all k, D x ϕ(v x ) ∈ Rϕ(x). Thus rank D x ψ < n if and only if there is v x ∈ R n \ {0} and t 1 , . . . , t n ∈ R n such that for all k, D x ϕ(v x ) = t k ϕ k (x). For all k and α k ∈ A k ,

D x ϕ k,α k = ϕ k,α k α T k .
Hence rank D x ψ < n if and only if there is v x ∈ R n \ {0} and t 1 , . . . , t n ∈ R n such that for all k and all α k ∈ A k , α T k v x = t k . In other words, rank D x ψ < n if and only if the A k are contained in parallel (affine) hyperplanes. If the affine span of n k=1 A k is R n , as we are assuming, then the latter is not possible and so D x ψ is injective, as desired. Now, assume that the affine span of n k=1 A k is not the entire R n . After a change of variables of the form x → exp(A log(x) + b), we can assume, without loss of generality, that the affine span of

n k=1 A k is R m × 0 ⊂ R n
with m < n. In this case we have that f is a random overdetermined system in the variables X 1 , . . . , X m . Moreover, arguing as before, we have that for all x ∈ R m , rank D x ψ = m. Hence, by Proposition 2.6, we have Z(f, R m + × {1}) = ∅ with probability one. This implies that if the affine span of n k=1 A k is not the entire R n we have ∅ = Z(f, R n + ) with probability one, and so Z(f, R n + ) = Z r (f, R n + ) with probability one.

Geometric Proof of Proposition 3.2. Without loss of generality, assume that the γ k are linearly independent. Otherwise the proposition is immediate, with both the integral and the expectation being zero. Note that the the left-hand side equality follows from Corollary 2.3, after taking ξ k (x) = exp(γ T k x + s k ) and observing that D x ξ k = exp(γ T k x + s k )γ T k . Hence we only need to bound the expectation of the number of real zeros.

Observe that the system (3.1) have a solution if and only if for all k, a k and b k have opposite signs. Moreover, in that case, the system is equivalent to the following linear system

γ t 1 X = -s 1 + ln(-b 1 /a 1 ) . . . γ t n X = -s n + ln(-b n /a n ) (3.11)
which has exactly one regular real solution. This system has this unique solution inside B if and only if the ln(-b k /a k ) are non-positive, which happens with probability 1/4 n . Hence

E h #Z r (h, B) = P(#Z r (h, B) = 0) = 1 4 n ,
as we wanted to show.

Analytical Proof of Proposition 3.2. The analytical proof is like the geometric one, but we compute directly now the left-hand side integral. Under the change of variables y k = -(γ T k x k + s k ), the left-hand side integral becomes

1 π n R n + n k=1 exp(-y k ) n k=1 (1 + exp(-2y k )) dy.
Under the further change of variables z k = exp(-y k ), this integral becomes

1 π n [0,1] n 1 n k=1 (1 + z 2 k ) dz.
Now, separating variables, this integral equals

1 π 1 0 1 1 + t 2 dt n = 1 4 n ,
where the last equality follows from arctan ′ (t) = 1/(1 + t 2 ).

Zeros of Random Unmixed Fewnomials

We now prove Theorem 1.5. We will actually prove a more general result, and then show that Theorem 1.5 is obtained as a corollary.

Theorem 4.1. Under the same notations and assumptions of Theorem 1.9, assume that

A = A 1 = • • • = A n and that the Minkowski sum n k=1 L(A, l k ) satisfies (MV) for some l : A → R, n k=1 L(A, l k ) = nL(A, l). Then E f #Z r (f, R n + ) ≤ n + 1 4 n V (L(A, l)) n + 1 (4.1)
where V (L(A, l)) ≤ #A is the number of vertices of L(A, l).

Proof of Theorem 4.1. The proof is the same as that of Theorem 1.9, but it differs in the combinatorics at two points: 1st point : When we write the decomposition (3.4), we only care about those α 1 , . . . , α n such that

n k=1 l k (α k ) α k is a vertex of n k=1 L(A, l k ).
However, due to the assumption (MV), we necessarily have that α 1 = • • • = α n , since those are the vertices of L(A, l). Hence we only need to look at polyhedral cells of the form M (α, . . . , α) where α ∈ A is a vertex of L(A, l)-and so nα a vertex of nL(A, l). Moreover, as we did in the proof of Theorem 1.9, we can assume, without loss of generality, that α = 0 and that for all k, l k (α) = 0.

2nd point : When we apply the Cauchy-Binet formula to obtain our decomposition. Since we use the same functions but different variances, we can write for each k, ϕ k = Σ k φ where φ(x) = x α α∈A and Σ k is a diagonal positive matrix with entries given by exp(l k (α)). It is important to note that the claim "we use the same function but different variances" really requires the extra assumption α = α 1 = • • • = α n . Otherwise, when doing the translation to put α 1 , . . . , α n at the origin, we will end up with ϕ 1 , . . . , ϕ n being different not only up to multiplication by a diagonal positive matrix. However, we do not need the l k (α) being independent of k, since turning them into zero only requires multiplying the ϕ k by positive diagonal matrices. Now,

   a T 1 D x ϕ 1 . . . a T n D x ϕ n    =    a T 1 . . . a T n       Σ 1 . . . Σ n    D x φ =    (Σ 1 a 1 ) T . . . (Σ n a n ) T    D x φ,
and so, by the Cauchy-Binet formula [18, Theorem 4.15] and the triangle inequality, det

   a T 1 D x ϕ 1 . . . a T n D x ϕ n    ≤ {β1,...,βn}⊂∈A\{0} n k=1 exp(l k (β k ))|a k,β k | det    D x φβ1 . . . D x φβn    ,
where the sum runs over subsets of size n. Hence we get t -1 n where the g k,α are independent continuous random variables. Then Z(g, P n R ) = ∅ with probability one.

Proof of Theorem 1.10. Consider an homogeneous fewnomial system f given by

f 1 = α∈A1 f 1,α X α . . . f q = α∈Aq f q,α X α
Then, by [38, (2.7)], Z(f, P n R ) admits a finite Whitney stratification. Thus, we can consider a partition of Z(f, P n R ) into disjoint subsets

Z 0 (f, P n R ), . . . , Z n (f, P n R ) ⊂ P n R such that that Z ℓ (f, P n R
) is an ℓ-dimensional smooth submanifold of Z(f, P n R ) or empty. Let H 1 , . . . , H n-q be random hyperplanes of P n R given by

l k = n i=0 a k,i X i
with the a k,i i.i.d. standard Gaussians-one can easily see that this is equivalent to moving by a random g ∈ O(n + 1) (with respect the Haar probability measure) a fixed (n -q)-plane L in P n R . By [START_REF] Bürgisser | Condition: The Geometry of Numerical Algorithms[END_REF]Proposition A.18], all the intersections

Z ℓ (f, P n R ) ∩ H 1 ∩ • • • ∩ H n-q (5.1)
are transversal with probability one. Hence, for ℓ < n -q, (5.1) is empty with probability one; and for ℓ ≥ n -q, (5.1) is, with probability one, either empty or a smooth submanifold of dimension ℓ -(n -q). By Poincaré's kinematic formula [21, Theorem A.55], we have that vol n-q Z n-q (f, P n R ) =

vol n-q P n-q R vol n P n R E H1,...,Hn-q #(Z n-q (f,

P n R ) ∩ H 1 ∩ • • • ∩ H n-q ).
Now, set the convention that vol n-q (Z(f, P n R )) = ∞, if for some ℓ > n -q, Z ℓ (f, P n R ) = ∅; and Z n-q (f, P n R ) = 0, if for all ℓ ≥ n -q, Z ℓ (f, P n R ) = ∅. Then, we have that vol n-q Z(f, P n R ) =

vol n-q P n-q R vol n P n R E H1,...,Hn-q #(Z(f,

P n R ) ∩ H 1 ∩ • • • ∩ H n-q ),
since if some for some ℓ > n -q, Z ℓ (f, P n R ) is non-empty, then the right-hand side will be infinite-(5.1) will be a non-empty positive dimensional smooth submanifold with probability one-and if for all ℓ ≥ n -q, Z ℓ (f, P n R ) is empty, then the right-hand side will be zero-(5.1) will be empty with probability one.

By the above discussion, we have that

E f vol n-q Z(f, P n R ) =
vol n-q P n-q R vol n P n R E f E H1,...,Hn-q #(Z(f,

P n R ) ∩ H 1 ∩ • • • ∩ H n-q ),
and so, by Tonelli's theorem, E f vol n-q Z(f, P n R ) =

vol n-q P n-q R vol n P n R E (f,l1,...,ln-q) #Z((f, l 1 , . . . , l n-q ), P n R ).

(5.2)

Assume (we prove this at the end) that, for all i, #Z((f, l 1 , . . . , l n-q ), H i ) = 0, where H i := Z(X i , P n R ) is the ith coordinate hyperplane in P n R , with probability one. Then Z((f, l 1 , . . . , l n-q ), P n R ) = {Z((f, l 1 , . . . , l n-q ), SP n + ) | S = diag(±1, . . . , ±1)}, where P n + := {x ∈ P n | x 0 > 0, . . . , x n > 0}, with probability one; and therefore, by symmetry, E f vol n-q Z(f, P n R ) = 2 n vol n-q P n-q R vol n P n R E f,l1,...,ln-q #Z((f, l 1 , . . . , l n-q ), P n + ).

(5.3) Now, by Theorem 1.1,

E f vol n-q Z(f, P n + ) ≤ 1 4 n
vol n-q P n-q R vol n P n R (n(n + 1)) n-q q k=1 t k (t k -1), since the l k have support of size n + 1, and we obtain the desired upper bound. We now prove our claim regarding coordinate hyperplanes. Without loss of generality assume that i = n. To show that Z((f, l 1 , . . . , l n-q ), H n ) is empty, it is enough to show that the random overdetermined system g := (f, l 1 , . . . , l n-q )(X 0 , . . . , X n-1 , 0), obtained by setting X n equal to 0, has no zeros in P n-1

R

. But this is precisely what Proposition 5.1 states.

Proof of Proposition 5.1. We will prove this by induction on n. The statement is obvious for n = 0. Consider the system g 1 (X 0 , . . . , X n-1 , 0), . . . , g n (X 0 , . . . , X n-1 , 0), then, by the induction hypothesis, this system does not have any zero in P n-1 R with probability one. In other words, as adding equations can only reduce the number of zeros, Z(g, P n R ) ∩ Z(X n , P n R ) = ∅ with probability one. Now, the same argument works, if instead of X n we consider X i . Therefore, for all i, Z(g, P n R ) ∩ Z(X i , P n R ) = ∅ with probability one. Now, we only need to show that {Z(g, SP n + ) | S = diag(±1, . . . , ±1)} = ∅, with probability one, where P n + := {x ∈ P n | x 0 > 0, . . . , x n > 0}. Now, by symmetry, it is enough to show that, with probability one, Z(g, P n + ) = ∅. However, the latter follows from the fact that a generic overdetermined system of Laurent polynomials does not have zeros in (C * ) n [76, Lemma 1]. This is not explicitly stated in [76, Lemma 1], but it can be easily seen by considering the overdetermined system of q > n equations in n variables as a system in q variables after adding q -n dummy variables-then, by [76, Lemma 1], the corresponding ideal is not propper for generic polynomials and, by evaluating to 1 the dummy variables, so it was not the original ideal.

(

  I2) ψ is injective. Moreover, we can further assume that (O) Ω = Ω. Under these assumptions, ψ(Ω) is an open embedded submanifold of n k=1 S m k . Now, consider the random hyperplanes

  and only if x ∈ Z(f, Ω). Moreover, by Sard's Theorem [59, §2] (cf.[START_REF] Bürgisser | Condition: The Geometry of Numerical Algorithms[END_REF] Proposition A.18]), the intersection

Curiously enough, this formula appears in the proof of many upper bounds in fewnomial theory[START_REF] Bihan | New fewnomial upper bounds from Gale dual polynomial systems[END_REF][START_REF] Bürgisser | Real Zeros of Mixed Random Fewnomial Systems[END_REF][START_REF] Bürgisser | On the number of real zeros of random fewnomials[END_REF].

The point of this appendix is to make clearer, for readers unfamiliar with measure theory, the measure-theoretic arguments underlying the proof of Theorem 2.1.Recall the following definitions for families of sets.

Acknowledgements

We thank Alicia Dickenstein for always making time to share her knowledge whenever we asked a question. We thank Maurice Rojas for inspiration and his enthusiasm for fewnomials. We are thankful to Peter Bürgisser and Elias Tsigaridas for helpful discussions.

A.E. and J.T.-C. thanks Paul Breiding, Sonja Petrovic and Gregory G. Smith for organizing the BIRS Workshop "Random Algebraic Geometry" (23w5070) at the Banff International Research Station, in Canada, from April 16 to April 21, 2023, which served as inspiration for this paper. A.E. is supported by NSF CCF 2110075. J.T.-C. thanks Evgeniya Lagoda for her constant support and Jazz G. Suchen for his insightful suggestion regarding equation (2.3). J.T.-C. also thanks Alperen A. Ergür, Eduardo Dueñez, José Iovino, José Morales, Nikos Salingaros, Chris La Valle, Chris Duffer and the UTSA Problem Solving Club (specially Thamara, Rachell, Adam, Taylor, Lila and Alyssa) for making him feel welcome at UTSA during his postdoctoral stay; Jeaheang Bang and Huan Xu for the nice atmosphere at the postdocs' office; and Alejandra Vincencio for making the official paperwork much easier during this stay. M.T. is funded by the European Union under the Grant Agreement no. 101044561, POSALG. Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or European Research Council (ERC). Neither the European Union nor ERC can be held responsible for them.

summands at most, instead of (t -1) n . Now, for each summand, the same computation as before gives us that each summand becomes

and so the proof ends as it did in the proof of Theorem 1.9.

The following lemma shows that Theorem 1.5 is a corollary of Theorem 4.1.

Lemma 4.2. Under the same notations and assumptions of Theorem 1.9, assume that

Then the following holds:

(1) If l = l 1 = . . . = l n then L(A,

If for all k and all α ∈ A we have l k (α) ≤ 0 with equality whenever α is a vertex of

and so taking Minkowski sums of K with itself is the same as taking integer dilations of K. This is essentially the first claim.

(2) For second claim, we have that, for all k, l k ≤ 0 and equality happens at the vertices of P . Thus, for all k, L(A, l k ) = (-∞, 0] × P . The rest follows as in [START_REF] Aliprantis | Infinite dimensional analysis: A hitchhiker's guide[END_REF].

We finish with a proof of Corollary 1.7.

Proof of Corollary 1.7. By Markov's inequality [21, Corollary 2.9] and Theorem 1.5,

) is a random variable with integer values. Now,

Hence the claim follows.

Volume of Random Projective Fewnomial Varieties

We prove Theorem 1.10, we will need the following proposition.

where {e 0 , . . . , e n } is the standard basis of R n+1 and ∆ n := conv(e 0 , . . . , e n ) the n-simplex. Consider the random overdetermined homogeneous fewnomial system g given by

Let X be a set and S ⊂ P(X) a non-empty collection of subsets of X. We say that:

(σ) [1, 4.1 Definition] S is a σ-algebra if S contains the empty set and it is closed under complements and countable pairwise disjoint unions, i.e.,

• ∅ ∈ S.

• for all A ∈ S, X \ A ∈ S.

• for every pairwise disjoint numerable subfamily {A n } n∈N of S, n∈N A n ∈ S.

(λ) [1, 4.9 Definition] S is a λ-system if S contains X, relative complements and monotone numerable unions, i.e.,

• X ∈ S.

• for all A, B ∈ S such that B ⊆ A, A \ B ∈ S.

• for every numerable subfamily {A n } n∈N of S that is increasing (for all n, A n ⊆ A n+1 ), [1, 4.9 Definition] S is a π-system if S is closed under finite intersections, i.e., for every A, B ∈ S, A ∩ B ∈ S.

Given any collection of sets, we can consider the σ-algebra, λ-system and π-system that it generates, by considering the smallest σ-algebra, λ-system or π-system that contains it. For a topological space, the following σ-algebra is essential.

Definition A.2. [1, 4.14 Definition] Let X be a topological space, its Borel σ-algebra, B(X), is the σ-algebra generated by the collection of its open subsets. The Dynkin's lemma allows us to extend statements from π-systems to the σ-algebra they generate by checking that they are satisfied at a λ-system. Theorem A.3 (Dynkin's lemma). [1, 4.11 Dynkin's Lemma] Let X be a set and S collection of subsets of X. If S is a π-system, then the λ-system it generates is a σ-algebra.

Recall that a topological space X is σ-compact if we can write X as a countable monotone union of compact sets and that a Borel measure of X is a measure of the form

The following theorem lies at the core of the proof of Theorem 2.1.

Theorem A.4. Let X be a σ-compact topological space and µ and ν Borel measures on X. Assume that there is a collection of open subsets U such that: (U0) U is a base for the topology of X, i.e., every open set in X is an union of open sets in U. Proof. By assumption, there is an increasing sequence of compact subsets {X n } n∈N such that X = n∈N X n . Assume without loss of generality that X 0 = ∅. For each n ∈ N, let

be the restriction of the Borel-measures µ and ν to X n . If for all n ∈ N, µ n = ν n , then µ = ν. To see this, take B ∈ B(X) and observe that

and

Fix arbitrary n ∈ N. We consider

which is a collection of open subsets of X n , and

If we show that U n satisfies (U0), (U1), (U3) and (U4) for µ n and ν n , then U n is a π-system contained in Λ n whose generated σ-algebra is B(X n ). If we show, moreover, that Λ n is a λ-system, then, by Dynkin's lemma (Theorem A.3),

and we are done, since then µ n = ν n .

We show now that U n satisfies (U0), (U1), (U3) and (U4) for µ n and ν n :

• U n satisfies (U0) and (U1), by the definition of the subspace topology and the construction of U n .

• For checking (U2) and (U3) for U n , we only need to write for U ∈ U,

This is possible, because U \ X n ∈ U, by (U1); µ(U ), ν(U ), µ(U \ X n ) and ν(U \ X n ) are finite, by (U2); and µ(U ) = ν(U ) and µ(U \ X n ) = ν(U \ X n ), by (U3).

We show now that Λ n is a λ-system:

• Since X n is compact and U n is an open cover, we have that there are U 1 , . . . , U ℓ ∈ U n such that

But then, by the inclusion-exclusion principle, (U2) and (U3),

Moreover, this argument shows that µ n (X n ) = ν(X n ) is finite, and so µ n (A) and ν n (A) are finite for all A ∈ B(X n ).

where the middle equality follows from A, B ∈ Λ n , since µ n (A), µ n (B), ν n (A) and ν n (B) are finite. Hence A \ B ∈ Λ n .

• Let {A k } k∈N ⊂ Λ n is an increasing family of subsets. Without loss of generality, assume that A 0 = ∅. Then By the previous paragraph and the assumption {A k } ⊂ Λ n , the right-hand sides are equal. Therefore the left-hand sides are so, and k∈N A k ∈ Λ n . Now, the proof is complete.


Ivan Magrin-Chagnolleau

Jean-Fran Cois Bonastre

? Fr

Ed Eric Bimbot
email: bimbot@sig.enst.fr?

EFFECT OF UTTERANCE DURATION AND PHONETIC CONTENT ON SPEAKER IDENTIFICATION USING SECOND-ORDER STATISTICAL METHODS

Second-order statistical methods show very good results for automatic speaker identi cation in controlled recording conditions 2]. These approaches are generally used on the entire speech material available. In this paper, we study the in uence of the content of the test speech material on the performances of such methods, i.e. under a more analytical approach 3 ] . The goal is to investigate on the kind of information which is used by these methods, and where it is located in the speech signal. Liquids and glides together, vowels, and more particularly nasal vowels and nasal consonants, are found to be particularly speaker speci c: test utterances of 1 second, composed in majority of acoustic material from one of these classes provide better speaker identi cation results than phonetically balanced test utterances, even though the training is done, in both cases, with 15 seconds of phonetically balanced speech. Nevertheless, results with other phoneme classes are never dramatically poor. These results tend to show that the speaker-dependent i nformation captured by long-term second-order statistics is consistently common to all phonetic classes, and that the homogeneity of the test material may i m p r o ve the quality of the estimates.

INTRODUCTION

In this paper, the in uence of phonetic content on the performance of a speaker identi cation system is investigated. Second-order statistical methods are chosen because they provide very good results with a low quantity of computations and with a restricted quantity o f s p e e c h material, provided recording conditions and channel distortions are controlled 2]. The goal of this work is to study how t h e performances of this family of approaches vary with the phonetic content of the test material. Several experiments were previously reported on the relative speaker discriminating properties of phonemes. In particular, Eatock et al. 5] used a VQ codebook based approach and concluded that nasals and vowels provided the best performances on an English language database. Le Floch e t a l . 9] investigated the properties of AR-vector models and concluded that vowels, diphtongs and nasals provided the best performances, also on an English language database. The speci city of the work reported in this paper comes from the fact that second-order statistical methods are used and that the training and test material are heterogeneous. The general experimental framework is the following: 15 seconds of training material coming from phonetically balanced sentences, are used to build a reference model for each s p e a k er. Then, speci c speech segments are selected from other phonetically balanced sentences in order to build a test pattern which is strongly biased towards a particular phoneme or phoneme class. The database used for our experiments contains 67 cooperative s p e a k ers recorded in a slightly noisy environment. For each s p e a k er, 50 French phonetically balanced sentences, i.e approximately 3 minutes of speech are recorded in a single session. This volume of speech allows to have a su cient number of occurences for most phonemes. The three second-order statistical measures used in the experiments are described in section 2.. Details on the speech database and on the signal analysis are given in section 3.. Section 4. reports preliminary experiments on utterance duration, and section 5. the experiments on the phonetic content. Finally, section 6. concludes on this work and gives some perspectives.

SPEAKER IDENTIFICATION MEASURES

The 3 speaker identi cation methods used in this work are inspired from statistical tests on covariance matrices 1], computed on acoustic parameters. The rst two speaker similarity measures are directly derived from maximum likelihood Gaussian classi ers 7]. The third one is based on a sphericity test between covariance matrices 8]. The measures, asymmetric in their original form, are symmetrised as a w eighted sum of the asymetric measure and its dual term. In previous work 2], this procedure was shown to improve performances. Let X and Y denote two c o variance matrices of a reference speaker and of a test speaker respectively, corresponding to the covariance of some spectral vectors computed along a sentence. Let x and y denote the means of the spectral vectors. Let M and N denote the number of spectral vectors used to estimate the covariance matrices and mean vectors, and p the dimension of the spectral vectors. The mathematical expression of the three measures that we used in our experiments, are given in Table 1. Note that, once the covariance matrices X and Y are inverted, and that their determinant i s e v aluated, the computation of the measures requires very few operations.

DATABASE AND SIGNAL ANALYSIS 3.1. Database

The corpus is composed of read phonetically balanced sentences 4], the phonetic transcription of which can be found in the BDSON database. The sentences are prompted on a screen. Recording begins and ends automatically using a speech activity detector. Each s e n tence is recorded through a SHURE SM10A microphon, and digitized at a 16 kHz

Signal Analysis

The speech analysis module extracts lterbank coe cients in the following way: a Winograd Fourier Transform is computed on Hamming windowed signal frames of 31.5 ms (i.e 504 samples) at a frame rate of 10 ms (160 samples). For each frame, spectral vectors of 24 Mel-Scale Triangular-Filter Bank coe cients are then calculated from the Fourier Transform power spectrum, and expressed in logarithmic scale. Covariance matrices and mean vectors are nally computed from these spectral vectors.

UTTERANCE DURATION

The rst set of experiments investigates on the in uence of utterance duration for second-order statistical methods.

PHONETIC CONTENT 5.1. Segmentation

In order to study the in uence of phonetic content on the performances of the second-order statistical methods, we used an automatic system for segmenting the speech material into speci c phonemes or phonetic classes. This system of automatic localisation is based on a bottom-up acousticphonetic decoder, which i s s p e a k er independent 3], 10], 6]. For each s e n tence, this decoder proposes a set of weighted phonetic hypotheses. These hypotheses are then aligned with the phonetic transcription of the sentence, by a leftright alignment algorithm. In order to obtain a high localisation accuracy, the algorithm is tuned with a high level of rejection for uncertain alignments: in this experiment, 55 % of the sentences were rejected. As a consequence, the phonetic events selected by this procedure can be considered as highly reliable, and quite typical in their category. The localisation algorithm gives a small kernel for a recognized phoneme. So the phoneme segments were extended to 5 frames before and 5 frames after the kernel. Therefore, the segments are not only composed with frames of the given phoneme, as they may a l s o include a small proportion of transitions.

Experimental Protocol

For this set of experiments, training material consists of all speech frames derived from 15 seconds of phonetically balanced sentences, i.e. no speci c phonetic events are selected. It is in fact the exact same material as the one used in the previous experiment on the in uence of utterance duration (see section 4.), with training durations of 15 seconds. For the tests, speci c phonetic classes and phonemes are selected on the rest of the speech material, by the procedure described in section 5.1.. For a particular phoneme, all the frames labeled as this given phoneme are concatenated together, and this material is divided into as many tests of 1 second as possible.

Description of the phoneme classes

Results for all phoneme classes and phonemes are not presented in this article: we c hose to report only on those for which more than 40 tests were carried out. A rst class, refered to as All , c o n tains all the phonemes. The experiments with test data from this class is however slightly di erent from the one in section 4., with 15 seconds for training and 1 second for testing: speech material in the class Allis composed of acoustic material clearly identi ed as a phoneme by the segmentation process, and in particular, it does not contain silences, pauses, or other non-linguistic events. The other classes are: Vowels (which c o n tains oral and nasal vowels but not glides), Oral Vowels, Nasal Vowels, Consonants (which contains also glides), Non-Nasal Consonants (which contains all the consonants except the nasal consonants), Nasal Consonants, Stop Consonants, Fricatives and Liquids+Glides (which form a single class).

Results

Results for various phoneme classes and individual phonemes are given in Table 3 and Table 4 respectively. The percentage of correct identi cations on all the tests is given, as well as the average of the correct identi cations per speaker. The number of tests is also indicated.

Comments

Quite surprisingly, measure G still performs better than Gc and S c : even though the mean vector within a phonetic class is expected to be strongly class dependent (and therefore not to match the training mean vector), it still keeps some consistence across phonetic classes. The results for the class Alloutperform slightly those for the 15 s 1 s experiment of section 4., probably because the speech material is, in the second case, more reliable. A second observation is that the results for each phoneme class is higher than the result of the class All . It is also the case for most of the phonemes. This tends to show that a phonetically homogeneous test material bene ts to the overall speaker identi cation performance, even though the training material does not share the same character. G are obtained with the phonemes /o/, /d/, /"/, and / ~ /, whereas /k/, /`/, /u/, /s/ and /y/ give the poorest performance levels.

CONCLUSION

Our experiments on the e ect of the phonetic content o n speaker identi cation using second-order statistics, tend to show that, on our database, the phonetic homogeneity o f the test material is usually a signi cant factor of improvement, even if the training material is heterogeneous. The phonetic classes that yield particularly good results are Liq-uids+Glides, Vowels (and more particularly Nasal Vowels), and Nasal Consonants, but results for other classes are never dramatically poor. There may therefore exist some kind of speaker-dependent tie between acoustic distributions across phonemes that is captured by the second-order statistical methods. This hypothesis has to be con rmed on other types of speech data, in particular noisy speech a n d non-contemporaneous recordings.

Table 1 .

 1 Expressions of the three symmetrized second-order statistical measures (\tr" denote the trace and \det" the determinant of a matrix). sampling frequency on 16 bits by a n O R OS AU22 board. The recording equipment was set up in the corridor of a university, i.e with a non-negligible background noise. The recordings are single-session. 67 speakers took part to the experiments, mostly students. They each recorded approximately 3 minutes of speech.

	ESCA EUROSPEECH'95.	337	ISSN 1018-4074

Table 2 .

 2 chosen to 1 second for two reasons. Firstly, if a longer duration is chosen, the number of tests for a given phoneme is less signi cant. Secondly, if the percentage of correct identi cations is too high, we felt that comparisons may l a c k statistical signi cance. Speaker identi cation: results in percentage of correct identi cations for di erent training and test durations. The numbers of tests for each test duration is indicated i n p arentheses.

	The best measure is always G, which con rms once again that the average of the spectral vectors is a signi cant source of speaker speci c information, for good quality con-temporaneous speech. Even though the measures are symmetric, a clear asym-metry of the results can be observed: for example, with a training of 10 seconds and a test of 2 seconds, the per-centage of correct identi cations with G is 95.3 %, while with a training of 2 seconds and a test of 10 seconds, it only reaches 88.4 %. In fact, if a training of 2 seconds is poorly representative of the speaker (for instance, if it con-tains a lot of silence), this has an e ect on all the tests utterances from this very speaker, which a ects consider-ably the overall score. If, conversely, a test of 2 seconds is

This allowed us to choose meaningful training and test durations for the second part of the work. Several durations for training and test are chosen: 15, 10, 6, 3 and 2 seconds for training 10, 6, 3, 2 and 1 second for testing. For each speaker, all sentences are randomly concatenated together. The silences at the beginning and the end of sentences are not removed, but they generally do not exceed 0.1 second. First, a certain amount of speech is selected for training (for example 15 seconds). Then, the rest is segmented in several test portions, each portion having a predetermined duration, until 20 test portions are obtained unless the speech material is exhausted. As a result of this experimental design, the experiments are text-independent. Percentages of correct identi cations are given in Table

2

. of poor quality, it only causes 1 mistake, which h a s little impact on the global performance. In the second part of our experiments, we c hose a 15 second training duration, which is a guaranty t o h a ve a reliable training and a su cient phonetic coverage. The test dura-tion is

Table 3 .

 3 In more detail, Vowels give better results than Consonants. Nasal Vowels outperform Vowels and Oral Vowels. Non-Nasal Consonants, and more particularly Stop Consonants or Fricatives, g i v e l o wer performances than all Consonants together, whereas Liquids+Glides and Nasal Consonants yield higher scores. Note that the class Liquids+Glides gives the best results altogether, except with S c for which t h e classes Nasal Consonants and Nasal Vowels perform better. In what concerns individual phonemes, the best scores with Speaker identi cation with speech material se-

	Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c	All (1334) 90.6 90.5 97.1 97.0 Vowels (1247) I M I M 80.8 80.6 92.3 92.1 83.5 83.4 92.0 91.7 Oral Vowels Nasal Vowels (1206) (262) I M I M 96.0 95.9 98.1 98.5 91.3 91.2 89.3 90.5 90.5 90.5 93.1 92.9 Consonants Non-Nasal (1247) Cons. (1186) I M I M 96.2 96.2 94.7 94.8 91.1 91.0 89.4 89.3 91.6 91.3 89.0 89.0 Nasal Cons. Stop Cons. (390) (693) I M I M 96.9 97.7 94.2 95.7 85.9 87.6 91.2 92.6 95.1 93.9 91.3 92.7 Fricatives Liquids + (486) Glides (277) I M I M 92.2 92.8 98.9 98.8 83.7 84.8 92.4 92.3 86.2 86.9 92.4 91.4
	lected from speci c phonetic classes. Training is composed of 15 s of phonetically balanced s p eech and test is composed of 1 s of phonetically biased s p eech. I = G l o b al correct iden-ti cation score, M = Average correct identi cation score over all speakers. The number of tests for each test con g-uration is indicated i n p arentheses.

Table 4 .

 4 Speaker identi cation with speech material selected f r om speci c phoneme realisations. Training is composed of 15 s of phonetically balanced s p eech and test is composed o f 1 s o f p h o n e t i c ally biased speech. I = G l o b al correct identi cation score, M = Average correct identi cation score over all speakers. The number of tests for each test con guration is indicated i n p arentheses.

	Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c Phonemes G Gc S c	(174) 92.0 92.2 96.2 97.5 (105) I M I M 79.9 81.3 86.7 88.0 81.0 82.5 87.6 88.9 /"/ /y/ (258) (66) I M I M 97.3 98.2 87.9 90.6 89.9 90.8 84.8 89.8 90.7 93.0 86.4 92.2 / / /a/ (329) (378) I M I M 96.7 97.1 95.8 95.7 91.8 91.8 87.8 86.9 91.2 90.9 90.7 89.3 /o/ /u/ (87) (75) I M I M 97.7 97.3 85.3 87.9 87.4 88.3 73.3 77.4 89.7 89.6 76.0 81.2 / ~ / /p/ (108) (113) I M I M 97.2 97.4 91.2 92.9 89.8 89.4 82.3 84.3 94.4 92.9 87.6 88.7 /t/ /k/ (196) (102) I M I M 90.8 92.3 82.4 86.0 86.7 87.1 78.4 84.4 89.3 90.4 78.4 81.0 /d/ /s/ (111) (194) I M I M 97.3 97.5 87.1 89.0 91.0 91.9 78.9 80.4 92.8 93.8 84.0 84.3 /v/ /`/ (42) (48) I M I M 90.5 89.2 83.3 82.9 76.2 77.0 77.1 76.0 76.2 75.7 75.0 73.6 /m/ /n/ (158) (149) I M I M 95.6 96.4 96.6 94.4 79.7 80.3 87.2 88.0 95.6 94.5 94.6 94.8