On Newton(like) inequalities for multivariate homogeneous polynomials

Let $p(x_1,...,x_m) = \sum_{r_1 + \cdots + r_m = n} a_{r_1,...,r_m} \prod_{1 \leq i \leq m } x_i^{r_{i}}$ be a homogeneous polynomial of degree $n$ in $m$ variables. We call such polynomial {\bf H-Stable} if $p(z_1,...,z_m) \neq 0$ provided that the real parts $Re(z_i) > 0: 1 \leq i \leq m$. It can be assumed WLOG that the coefficients $a_{r_1,...,r_m} := a_{R} \geq 0$.\\ This notion from {\it Control Theory} is closely related to the notion of {\it Hyperbolicity} intensively used in the {\it PDE} theory.\\ Let $R_0; R_1,...,R_k$ are integer vectors and $R_0 = \sum_{1 \leq j \leq k} a_j R_j$, where the real numbers $a_j \geq 0: 1 \leq i \leq k$ and $\sum_{1 \leq j \leq k} a_j = 1$. We define, for an integer vector $R = (r_1,...,r_m)$, $R! =: \prod_{1 \leq i \leq m } r_{i}!$.\\ We prove that $\log(a_{R} R!) \geq \sum_{1 \leq j \leq k} a_j \log(a_{R} R_{j}!) - n \alpha_n$, where $\frac{1}{2} \log(2) \leq \alpha_n \leq \log(\frac{n^n}{n!})$ and get better bounds on $\alpha_n$ for sparse polynomials. We relax a notion of {\bf 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 for {\bf H-Stability} and can be used for the identification of multivariate stable linear system, i.e. for the interpolation of {\bf H-Stable} polynomials.


MTNS-2008 accepted paper



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