On multivariate Newton-like inequalities

TL;DR

This paper introduces a class of multivariate entire functions called Strongly Log-Concave, generalizes Newton inequalities to this class, and shows their support sets are discretely convex, with proofs based on convex relaxations and bounds related to the Van Der Waerden inequality.

Contribution

It defines Strongly Log-Concave functions, extends Newton inequalities to multivariate cases, and links support convexity to these inequalities, providing new theoretical insights.

Findings

Strongly Log-Concave functions generalize Minkowski volume polynomials.

Multivariate Newton-like inequalities are established for this class.

Support of such functions is proven to be discretely convex.

Abstract

We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$ variables is called {\bf 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 (via {\it propagation}) on the standard univarite (or homogeneous bivariate) {\bf Newton Inequlities}. We prove analogues of (univariate) {\bf Newton Inequlities} in the (multivariate) {\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like) inequalities is the fact that the support $supp(f)$ of a {\bf Strongly Log-Concave} entire function $f$ is discretely 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 {\bf Van Der Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.