多重对称多项式正性的判定
Deciding positivity of multisymmetric polynomials
摘要：
如何证明一个多项式函数的非负性是实代数的核心问题，在最优化中也有重要的应用。在这篇文章中，我们调查的非负性问题的背景下，多对称多项式。在这种情况下，我们一般化的非负对称多项式的特点，通过调整的方法证明第二作者。一个特殊的情况下，我们的结果可以应用的问题是证明（多）对称多项式定义一个凸函数。作为我们的主要结果的直接推论，我们能够推导出，在（多）对称多项式的情况下，一个固定的程度测试凸性可以在一个时间，这是多项式的变量的数量。这与一般情况形成鲜明对比，在一般情况下，已知在四次多项式的情况下，凸性测试已经是NP难的。
英文摘要：
The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of multisymmetric polynomials. In this setting we generalize the characterization of non-negative symmetric polynomials by adapting the method of proof developed by the second author. One particular case where our results can be applied is the question of certifying that a (multi-)symmetric polynomial defines a convex function. As a direct corollary of our main result we are able to derive that in the case of (multi-)symmetric polynomials of a fixed degree testing for convexity can be done in a time which is polynomial in the number of variables. This is in sharp contrast to the general case, where it is known that testing for convexity is NP-hard already in the case of quartic polynomials.
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