用双曲优化证明多项式的非负性

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED
J. Saunderson
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引用次数: 15

摘要

通过求解双曲优化问题——一类推广半定规划的凸优化问题,给出了证明多元多项式全局非负性的新方法。我们展示了如何从任何双曲多项式生成非负多项式族(我们称之为非负多项式的双曲证明)。我们研究了在$n$变量中有一个次为$d$的双曲多项式使得相关的非负双曲证明不是平方和的对$(n,d)$。如果$d\geq 4$,我们显示每当$n\geq 4$。在三次情况下,我们在$43$变量中找到一个显式双曲三次,它给出了不是平方和的双曲证明。作为一个推论,我们得到了已知的第一个没有幂的双曲三次方程具有确定的行列式表示。我们的方法还允许我们证明,给定一个三次$p$和一个方向$e$,决策问题“$p$相对于$e$是双曲的吗?”是协同np困难的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Certifying Polynomial Nonnegativity via Hyperbolic Optimization
We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs $(n,d)$ for which there is a hyperbolic polynomial of degree $d$ in $n$ variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If $d\geq 4$ we show that this occurs whenever $n\geq 4$. In the degree three case, we find an explicit hyperbolic cubic in $43$ variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic $p$, and a direction $e$, the decision problem "Is $p$ hyperbolic with respect to $e$?" is co-NP hard.
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CiteScore
2.20
自引率
0.00%
发文量
19
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