On Newton(like) inequalities for multivariate homogeneous polynomials

Let p(x1, ..., xm) = ∑ r1+···+rm=n ar1,...,rm ∏ 1≤i≤m x ri i be a homogeneous polynomial of degree n in m variables. We call such polynomial H-Stable if p(z1, ..., zm) 6= 0 provided that the real parts Re(zi) > 0 : 1 ≤ i ≤ m. It can be assumed WLOG that the coefficients ar1,...,rm := aR ≥ 0. This notion from Control Theory is closely related to the notion of Hyperbolicity intensively used in the PDE theory. Let R0;R1, ..., Rk are integer vectors and R0 = ∑ 1≤j≤k ajRj , where the real numbers aj ≥ 0 : 1 ≤ i ≤ k and ∑ 1≤j≤k aj = 1. We define, for an integer vector R = (r1, ..., rm), R! =: ∏ 1≤i≤m ri!. We prove that log(aRR!) ≥ ∑ 1≤j≤k aj log(aRRj !) − nαn, where 1 2 log(2) ≤ αn ≤ log( n n! ) and get better bounds on αn for sparse polynomials. We relax a notion of H-Stability by introducing two classes of homogeneous polynomials: Alexandrov-Fenchel polynomials and Strongly Log-Concave polynomials, prove analogous inequalities for those classes and use them to prove L-convexity of the supports of polynomials from those classes. We also present a new view on the standard, i.e. when m = 2, Newton inequalities and pose some open problems. Our results provide new necessary conditions forH-Stability and can be used for the identification of multivariate stable linear system, i.e. for the interpolation of H-Stable polynomials. 1 Standard Newton Inequalities Definition 1.1: We define the following closed subset of Rn+1 : LC = {(d0, ..., dn) : di ≥ 0, 0 ≤ i ≤ n; di ≥ di−1di+1, 1 ≤ i ≤ n− 1. We also define a weighted shift operator Shiftc : Rn+1 −→ Rn+1, Shiftc((x0, ..., xn) ) = (c0x1, ..., cn−1xn, 0) . ∗[email protected]. Los Alamos National Laboratory, Los Alamos, NM.