On multivariate Newton-like inequalities

Abstract

Abstract 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. KeywordsEntire FunctionConvex ConeMixed VolumeInteger VectorConvex Compact SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.