On multivariate Newton-like inequalities

By Leonid Gurvits. 2008

57 citations recorded by the source corpus.

We study multivariate entire functions and polynomials with non-negative coefficients. A class of Strongly Log-Concave entire functions, generalizing Minkowski volume polynomials, is introduced: an entire function f in m variables is called Strongly Log-Concave if the function \((\partial x_{1})^{c_{1}}...(\partial x_{m})^{c_{m}}f\) is either zero or \(\log((\partial x_{1})^{c_{1}}...(\partial x_{m})^{c_{m}}f)\) is concave on \(R_{+}^{m}\) . We start with yet another point of view (of propagation) on the standard univariate (or homogeneous bivariate) Newton Inequalities. We prove analogues of the Newton Inequalities in the multivariate Strongly Log-Concave case. One of the corollaries of our new Newton-like inequalities is the fact that the support supp(f) of a Strongly Log-Concave entire function f is pseudo-convex (D-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives Der f (r 1,...,r m ) of f at zero and on the lower bounds on Der f (r 1,...,r m ), which generalize the van der Waerden-Egorychev-Falikman inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0e8a81f57b1b820ec0d54b961b60310a93240614

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In Applied Mathematics

June Huh

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Logarithmic concavity of Schur and related polynomials (2019, background): “ and only if the sequence a 0;:::;a dhas no internal zeros 8 and a2 k d k 2 ¥ a k 1 d k 1 a k 1 k 1 for all 0 €k d. Polynomials satisfying the second condition of Deﬁnition5, introduced by Gurvits in [Gur09], are called strongly log-concave. See [BH19, Section 5] for a proof of the equivalence of th”