3 research outputs found

    Real tropicalization and negative faces of the Newton polytope

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    In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1<0,,fk<0}S = \{ f_1 < 0, \dots , f_k < 0\} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,,fkf_1, \dots, f_k. We describe a cone that depends only on the face structure of the Newton polytopes of f1,,fkf_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}S = \{ f < 0\} and the polynomial ff has generic coefficients. Furthermore, we show that for a maximally sparse polynomial ff the real tropicalization of S={f<0}S = \{ f < 0\} is determined by the outer normal cones of the Newton polytope of ff and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant

    On the Number of Real Zeros of Random Sparse Polynomial Systems

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    Consider a random system f1(x)=0,,fn(x)=0\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0 of nn random real polynomials in nn variables, where each fk\mathfrak{f}_k has a prescribed set of exponent vectors in a set AkZnA_k\subseteq \mathbb{Z}^n of size tkt_k. Assuming that the coefficients of the fk\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 4nk=1ntk(tk1)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

    On generalizing Descartes' rule of signs to hypersurfaces

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    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
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