  


On Newton(like) inequalities for multivariate homogeneous polynomials
Leonid Gurvits(gurvitslanl.gov) Abstract: 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 HStable} 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 HStability} by introducing two classes of homogeneous polynomials: AlexandrovFenchel polynomials and Strongly LogConcave 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 HStability} and can be used for the identification of multivariate stable linear system, i.e. for the interpolation of {\bf HStable} polynomials. Keywords: polynomial, logconcavity, matroid Category 1: Combinatorial Optimization Citation: MTNS2008 accepted paper Download: [PDF] Entry Submitted: 06/04/2008 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  