On multivariate Newton-like inequalities
نویسنده
چکیده
We study multivariate entire functions and polynomials with non-negative coefficients. A class of Strongly Log-Concave entire functions, generalizingMinkowski volume polynomials, is introduced: an entire function f in m variables is called Strongly Log-Concave if the function (∂x1) c1 ...(∂xm) mf is either zero or log((∂x1) c1 ...(∂xm) mf) is concave on R + . We start with yet another point of view (of propagation) on the standard univarite (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 discretely convex (D-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives Derf (r1, ..., rm) of f at zero and on the lower bounds on Derf (r1, ..., rm), which generalize the Van Der Waerden-Falikman-Egorychev inequality for the permanent of doublystochastic matrices. A few open questions are posed in the final section. ∗ [email protected]. Los Alamos National Laboratory, Los Alamos, NM. 1
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