Bounds for absolute positiveness of multivariate polynomials
Let \(P(x_1,\dots, x_n)\) be a multivariate polynomial with real coefficients. A real number \(B\) is a bound for the absolute positiveness of \(P\) if for every \(x_1,\dots, x_n\geq B\), \(P\) and all of its non-zero partial derivatives of every order are positive. The problem of computing bounds, if any, for a multivariate polynomial \(P\) is treated in this paper. In an earlier work, \textit{H. Hong} and \textit{D. Jakus} [Testing positiveness of polynomials, Technical Report 96-02, RISC-Linz, Johannes Kepler University, Linz, Austria (1996)] proved that the existence of a bound for the absolute positiveness of \(P\) is equivalent to the property: every dominating monomial of \(P\) has positive coefficients. For such a polynomial \(P= \sum_{\mu\in I}a_\mu x^\mu\in \mathbb{R}[x]\) a bound is given by the following formula (Theorem 2.2) \[ B_p= {1\over\Omega_n} \max_{a_\mu< 0} \min_{a_\mu> 0,\nu>\mu} \Biggl|{a_\mu\over a_\nu}\Biggr| ^{{1\over| \nu-\mu|}},\tag{1} \] where \(\Omega_n= 1- \root n\of{1/2}\), \(\mu:= (\mu_1,\dots, \mu_n)\in \mathbb{N}^n\), and \(|\mu|:= \mu_1+\cdots+ \mu_n\). The quality of (1) is remarked by the following estimate (Theorem 2.3) \[ {B_p\over A_p}\leq {1\over\Omega_n} {d_1+\cdots+ d_n\over \ln(2)}, \] where \(d_i= \deg_{x_i}P\), and \(A_p\) is the infimum of all the bounds for the absolute positiveness of the polynomial \(P\). The last section of the paper is devoted to compare the bound \(B_p\) given in (1), with the known bounds for the real roots of univariate polynomials.
scientific article; zbMATH DE number 1204289