Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set $S = \{ f_1 < 0, \dots , f_k < 0\}$ defined in the positive orthant and the combinatorial properties of the defining polynomials $f_1, \dots, f_k$. We describe a cone that depends only on the face structure of the Newton polytopes of $f_1, \dots ,f_k$ and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if $S = \{ f < 0\}$ and the polynomial $f$ has generic coefficients. Furthermore, we show that for a maximally sparse polynomial $f$ the real tropicalization of $S = \{ f < 0\}$ is determined by the outer normal cones of the Newton polytope of $f$ and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant

Binary geometries from pellytopes

Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell\u27s numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron

Viro's patchworking and the signed reduced A-discriminant

Computing the isotopy type of a hypersurface, defined as the positive real zero set of a multivariate polynomial, is a challenging problem in real algebraic geometry. We focus on the case where the defining polynomial has combinatorially restricted exponent vectors and fixed coefficient signs, enabling faster computation of the isotopy type. In particular, Viro's patchworking provides a polyhedral complex that has the same isotopy type as the hypersurface, for certain choices of the coefficients. So we present properties of the signed support, focussing mainly on the case of n-variate (n+3)-nomials, that ensure all possible isotopy types can be obtained via patchworking. To prove this, we study the signed reduced A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions

Geometry of the signed support of a multivariate polynomial and Descartes\u27 rule of signs

We investigate the signed support, that is, the set of the exponent vectors and the signs of the coefficients, of a multivariate polynomial $f$. We describe conditions on the signed support ensuring that the semi-algebraic set, denoted as $\{ f < 0 \}$, containing points in the positive real orthant where $f$ takes negative values, has at most one connected component. These results generalize Descartes\u27 rule of signs in the sense that they provide a bound which is independent of the values of the coefficients and the degree of the polynomial. Based on how the exponent vectors lie on the faces of the Newton polytope, we give a recursive algorithm that verifies a sufficient condition for the set $\{ f < 0 \}$ to have one connected component. We apply the algorithm to reaction networks in order to prove that the parameter region of multistationarity of a ubiquitous network comprising phosphorylation cycles is connected.Final version to appear in SIAM Journal on Applied Algebra and Geometr

On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Assuming that the coefficients of the $\mathfrak{f}_k$ 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 $4^{-n} \prod_{k=1}^n t_k(t_k-1)$. 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.Comment: 26 pages. Different original titl

Determination of Condensed Tannins in Tropical Legume Forages

The effect of condensed tannins (proanthocyanidins) in ruminant nutrition is complex and needs further interdiscipli­nary reinvestigations with improved analytical methods. With the « cell rupture » theory of legume pasture bloat (Howarth et al., 1978), the role of condensed tannins seems to be secondary; they are acting as protein precipitants (Goplen et al., 1980) and inhibiting microbiological proliferation in cell­ular spaces (Howarth et al., 1982). Their assumed interference with digesting enzymes in vivo is doubtful (Mole and Water­man, 1985). On the importance of condensed tannins in nutrition, several reviews were published (McLeod, 1974; Tempel, 1982; Kumar and Singh, 1984 and Mehansho et al., 1987). Several methods have been suggested for the determination of condensed tannins, but none are problem-free. A new, sim­ple, and specific method for the determination of soluble and insoluble condensed tannins in fresh leaf samples follows

Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1&lt;0,…,fk&lt;0} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,…,fk. We describe a cone that depends only on the face structure of the Newton polytopes of f1,…,fk and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f&lt;0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f&lt;0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.</p

Geometry of the Signed Support of a Multivariate Polynomial and Descartes’ Rule of Signs

We investigate the signed support, that is, the set of the exponent vectors and the signs of the coefficients, of a multivariate polynomial ƒ We describe conditions on the signed support ensuring that the semialgebraic set, denoted as [f &lt; 0}, containing points in the positive real orthant where takes negative values, has at most one connected component. These results generalize Descartes’ rule of signs in the sense that they provide a bound which is independent of the values of the coefficients and the degree of the polynomial. Based on how the exponent vectors lie on the faces of the Newton polytope, we give a recursive algorithm that verifies a sufficient condition for the set [f &lt; 0} to have one connected component. We apply the algorithm to reaction networks in order to prove that the parameter region of multistationarity of a ubiquitous network comprising phosphorylation cycles is connected

Signed Support of Multivariate Polynomials and Applications

This thesis includes six papers that investigate three different areas: chemical reaction network theory, Descartes’ rule of signs, and real tropicalization. A common thread among them is the significant role played by the signed support of multivariate polynomials.Paper I and II focus on chemical reaction networks. In Paper I, we describe a general algorithm for verifying connectivity of the parameter region of multistationarity of a reaction network and apply it to several biologically relevant networks. In Paper II, our focus is on two families of phosphorylation networks, called n-site phosphorylation networks. We provide a proof showing that their parameter region of multistationarity is connected for every n ∈ N≥2.In Paper III and IV, we present combinatorial conditions on the signed support that provide upper bounds on the number of connected components of the set in the positive real orthant where the polynomial takes negative values. We frame this problem as a generalization of Descartes’ rule of signs to multivariate polynomials. The methods developed in Paper III and IV are crucial for the arguments used in Paper I and II.In Paper V, we investigate the real tropicalization of semi-algebraic sets and show its relation to the signed support of the polynomials defining these sets. In Paper VI, we study the signed A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions. In such cases, Viro’s patchworking becomes applicable for determining all isotopy types of hypersurfaces in the positive real orthant with a prescribed signed support for their defining polynomials

On generalizing Descartes' rule of signs to hypersurfaces

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.Comment: Final version to appear in Advances in Mathematic