On the complexity of the generalized MinRank problem

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• 作者：Jean-Charles Faug&egrave ; re^a ; ^{Jean-Charles.Faugere@inria.fr} ; Mohab Safey El Din^a ; ^c ; ^{Mohab.Safey@lip6.fr} ; Pierre-Jean Spaenlehauer^b ; ^a ; ^{pierre-jean.spaenlehauer@m4x.org}
• 关键词：MinRank ; Grö ; bner basis ; Determinantal ; Bi-homogeneous ; Structured algebraic systems
• 刊名：Journal of Symbolic Computation
• 出版年：2013
• 出版时间：August, 2013
• 年：2013
• 卷：55
• 期：Complete
• 页码：30-58
• 全文大小：456 K

文摘

We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most r. A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size of the matrix. We give new complexity bounds for solving this problem using Gr?bner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree . We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.