Bounds on absolute positiveness of multivariate polynomials
A multivariate polynomial \(P(x_1,\dots,x_n)\) with real coefficients is said to be absolutely positive from a real value \(B\) if \(P\) and all its non-zero partial derivatives of arbitrary order are positive for \(x_1 \geq B,\ldots,x_n \geq B\). The authors present a framework for obtaining bounds \(B\) on absolute positiveness of \(P\). The structure of \(P\) is governed by the signs of the coefficients and a special coefficient weight matrix \(\Delta\) with certain constraints on its entries. The bound of [\textit{H. Hong}, J. Symb. Comput. 25, No.~5, 571--585 (1998; Zbl 0913.65013)] is obtained for a special choice of \(\Delta\), and it is shown to be at most a multiplicative factor of \(O(n)\) from the best possible bound within this framework. Moreover, the overestimation factor (i.e., the ratio to the infimum of all bounds on absolute positiveness of \(P\)) is estimated for any \(B_\Delta\), and an iterative way to improve it is given. In the univariate case, the authors derive the minimum number of weights necessary to obtain a bound within the framework with limited overestimation.
scientific article; zbMATH DE number 5710824
scientific article