正维多项式理想的几何对合基与SDP方法
Geometric involutive bases for positive dimensional polynomial ideals and SDP methods
摘要：
多项式方程组的几何对合基起源于几何学家Cartan和Kuranishi对偏微分方程组的延拓和投影方法。它们对于理想隶属度的数值检验和多项式系统的求解是有用的。在本文中，我们进一步发展我们的符号数值方法，这样的基地。我们给出了显式提取和减少中间系统和输出基的次数的方法。本文还讨论了正维理想对合性判据的数值计算算法。我们也受到了Lasserre和合作者最近的一些出色工作的激励，他们将我们的延拓投影对合准则作为他们半定基规划（SDP）方法的一部分，用于确定零维多项式理想的实根。因此，在本文中，我们开始探索几何对合基和这些方法之间的相互作用，特别是在正维的情况下。出于这些方法的推广，正维的情况下，我们探讨几何对合基地和新的SDP方法之间的相互作用。
英文摘要：
Geometric involutive bases for polynomial systems of equations have their origin in the prolongation and projection methods of the geometers Cartan and Kuranishi for systems of PDE. They are useful for numerical ideal membership testing and the solution of polynomial systems. In this paper we further develop our symbolic-numeric methods for such bases. We give methods to explicitly extract and decrease the degree of intermediate systems and the output basis. Algorithms for the numerical computation of involutivity criteria for positive dimensional ideals are also discussed. We were also motivated by some remarkable recent work by Lasserre and collaborators who employed our prolongation projection involutive criteria as a part of their semi-definite based programming (SDP) method for identifying the real radical of zero dimensional polynomial ideals. Consequently in this paper we begin an exploration of the interaction between geometric involutive bases and these methods particularly in the positive dimensional case. Motivated by the extension of these methods to the positive dimensional case we explore the interplay between geometric involutive bases and the new SDP methods.
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