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Frédéric Bihan

Publications and source records attributed to Frédéric Bihan.

18 recordsLinked to original sources
arXiv2026

Descartes' rule of signs for arbitrary fewnomial systems

The classical Descartes' rule of signs bounds the number of positive roots of a real polynomial in one variable by the number of sign changes of its coefficients. This bound only depends on the order of the exponents and on the signs of the coefficients, that is, on the oriented matroids of the exponent and coefficient matrices. We give a multivariate generalization of this rule to arbitrary fewnomial systems: for systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials, we obtain upper bounds for the number of nondegenerate positive solutions which only depend on sign conditions on the maximal minors of the coefficient matrix and of the exponent matrix and its liftings. For $k=1$ they are the Descartes rules of signs for circuits of Bihan-Dickenstein and Bihan-Dickenstein-Forsgård. Our approach starts from the Gale duality and the Khovanskii-Rolle argument of Bihan and Sottile, which reduce the problem to counting common zeros of iterated Jacobians in a polyhedron $Δ$. This polyhedron is a chamber of a hyperplane arrangement whose combinatorics is determined by the oriented matroid of the coefficient matrix. We bound the number of these zeros by the Bézout number minus the zeros forced in the other chambers by a topological degree, which we compute by a facet-count formula involving the signs of minors of a Gale dual matrix of the exponent matrix. This leads to a recursion over the flats of the arrangement, which ends on lines with the Descartes rules for circuits. In this way the coefficients enter the bounds through the combinatorics of the arrangement, and the exponents through sign conditions on minors.

math.CO↗
arXiv2025

Sparse systems with high local multiplicity

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

math.AG↗
arXiv2024

New bounds for the number of connected components of fewnomial hypersurfaces

We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors.

math.AG↗
arXiv2022

Optimal Descartes' Rule of Signs for Circuits

We present an optimal version of Descartes' rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with n+2 monomials. This sharp upper bound is given in terms of the sign variation of a sequence associated to the exponents and the coefficients of the system.

math.AG↗
arXiv2018

A Polyhedral Method for Sparse Systems with many Positive Solutions

We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we get new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property.

math.CO↗
arXiv2018

Regions of multistationarity in cascades of Goldbeter-Koshland loops

We consider cascades of enzymatic Goldbeter-Koshland loops with any number n of layers, for which there exist two layers involving the same phosphatase. Even if the number of variables and the number of conservation laws grow linearly with n, we find explicit regions in reaction rate constant and total conservation constant space for which the associated mass-action kinetics dynamical system is multistationary. Our computations are based on the theoretical results of our companion paper in arXiv:1807.05157, which are inspired by results in real algebraic geometry by Bihan, Santos and Spaenlehauer.

q-bio.MN↗
arXiv2018

Lower bounds for positive roots and regions of multistationarity in chemical reaction networks

Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system has more than one positive solution, based on the recent article by Bihan, Santos and Spaenlehauer. We apply this approach to find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary.

math.DS↗
arXiv2017

Criteria for strict monotonicity of the mixed volume of convex polytopes

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes $P_1,\dots,P_n$ whose number of isolated solutions equals the normalized volume of the convex hull of $P_1\cup\dots\cup P_n$. In addition, we obtain an analog of Cramer's rule for sparse polynomial systems.

math.MG↗
arXiv2016

Descartes' Rule of Signs for Polynomial Systems supported on Circuits

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit.

math.AG↗
arXiv2016

Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes

Consider a regular triangulation of the convex-hull $P$ of a set $\mathcal A$ of $n$ points in $\mathbb R^d$, and a real matrix $C$ of size $d \times n$. A version of Viro's method allows to construct from these data an unmixed polynomial system with support $\mathcal A$ and coefficient matrix $C$ whose number of positive solutions is bounded from below by the number of $d$-simplices which are positively decorated by $C$. We show that all the $d$-simplices of a triangulation can be positively decorated if and only if the triangulation is balanced, which in turn is equivalent to the fact that its dual graph is bipartite. This allows us to identify, among classical families, monomial supports which admit maximally positive systems, i.e. systems all toric complex solutions of which are real and positive. These families give some evidence in favor of a conjecture due to Bihan. We also use this technique in order to construct fewnomial systems with many positive solutions. This is done by considering a simplicial complex with bipartite dual graph included in a regular triangulation of the cyclic polytope.

math.AG↗
arXiv2014

Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems

Given convex polytopes $P_1,...,P_r$ in $R^n$ and finite subsets $W_I$ of the Minkowsky sums $P_I=\sum_{i \in I} P_i$, we consider the quantity $N(W)=\sum_{I \subset {\bf [}r {\bf ]}} {(-1)}^{r-|I|} \big| W_I \big|$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(W)$ under some assumptions on the family $W=(W_I)$. In particular, we are able to show the nonnegativity of $N(W)$ in some important cases. The quantity $N(W)$ associated with the family defined by $W_I=\sum_{i \in I} W_i$ is called discrete mixed volume of $W_1,...,W_r$. We show that for $r=n$ the discrete mixed volume provides an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $W_1,...,W_n$. We also prove that the discrete mixed volume of $W_1,...,W_r$ is bounded from above by the Kouchnirenko number $\prod_{i=1}^r (|W_i|-1)$. For $r=n$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $W_1,...,W_n$. This conjecture was disproved, but our result shows that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.

math.AG↗
arXiv2007

Bounds on the number of real solutions to polynomial equations

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.

math.AG↗
arXiv2007

Gale duality for complete intersections

We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections.

math.AG↗
arXiv2005

Is every toric variety an M-variety?

A complex algebraic variety X defined over the real numbers is called an M-variety if the sum of its Betti numbers (for homology with closed supports and coefficients in Z/2) coincides with the corresponding sum for the real part of X. It has been known for a long time that any nonsingular complete toric variety is an M-variety. In this paper we consider whether this remains true for toric varieties that are singular or not complete, and we give a positive answer when the dimension of X is less than or equal to 3.

math.AG↗
arXiv2003

Asymptotiques de nombres de Betti d'hypersurfaces projectives réelles

We are interested in the maximal values of the Betti numbers b_i({\mathbb R}X_m^n) for fixed i,m,n; where {\mathbb R}X_m^n is the real part of a real nonsingular hypersurface of degree m in the complex projective space {\mathbb C}P^n, and in the maximal values of the Betti numbers b_i({\mathbb R}Y_{2k}^n) for fixed i,k,n; where {\mathbb R}Y_{2k}^n is the real part of a double covering Y_{2k}^n of{\mathbb C}P^n ramified over some real nonsingular hypersurface of degree 2k. We show the existence of limits lim_{m -> +\infty} [Max b_i({\mathbb R}X_m^n)]/m^n=h_{i,n} and lim_{k -> +\infty} [Max b_i({\mathbb R}Y_{2k}^n)]/k^n =d_{i,n}. We construct real nonsingular hypersurfaces as small perturbations of double hypersurfaces using the Viro method. This construction enables us to obtain recursive lower bounds for the h_{0,n} and d_{0,n}, and inequalities h_{0,3} >= d_{0,2}/6+1/12, h_{1,3} >= d_{1,2}/6+1/6. As applications, we show the existence, for any integer n >= 5, of real algebraic hypersurfaces in {\mathbb C}P^n which are not T-hypersurfaces, and we prove inequalities 35/96 <= h_{0,3} <= 5/12, 35/48 <= h_{1,3} <= 5/6.

math.AG↗