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TitleOn the complexity of the generalized MinRank problem
Author(s) Jean-Charles Faugère, Mohab Safey, Pierre-Jean Spaenlehauer
TypeArticle in Journal
AbstractAbstract 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 r + 1 of the matrix. We give new complexity bounds for solving this problem using Gröbner 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 ( D , 1 ) . We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.
KeywordsMinRank, Gröbner basis, Determinantal, Bi-homogeneous, Structured algebraic systems
URL http://www.sciencedirect.com/science/article/pii/S0747717113000485
JournalJournal of Symbolic Computation
Pages30 - 58
Translation No
Refereed No