On multivariate Newton-like inequalities

Leonid Gurvits ()
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Leonid Gurvits: Los Alamos National Laboratory

Chapter Chapter 4 in Advances in Combinatorial Mathematics, 2009, pp 61-78 from Springer

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.

Keywords: Entire Function; Convex Cone; Mixed Volume; Integer Vector; Convex Compact Subset (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-03562-3_4

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DOI: 10.1007/978-3-642-03562-3_4

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