3 research outputs found
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 defined in the positive
orthant and the combinatorial properties of the defining polynomials . We describe a cone that depends only on the face structure of the
Newton polytopes of 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 and the polynomial has generic coefficients. Furthermore, we show that
for a maximally sparse polynomial the real tropicalization of is determined by the outer normal cones of the Newton polytope of 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
Consider a random system of
random real polynomials in variables, where each has a
prescribed set of exponent vectors in a set of size
. Assuming that the coefficients of the 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 . 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
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