Refined F_5 Algorithms for Ideals of Minors of Square Matrices
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We consider the problem of computing a grevlex Gröbner basis for the set F_r(M) of minors of size r of an n× n matrix M of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular sequences; in fact, the ideal ⟨ F_r(M) ⟩ cannot be generated by a regular sequence. As such, when using the general-purpose algorithm F_5 to find the sought Gröbner basis, some computing time is wasted on reductions to zero. We use known results about the first syzygy module of F_r(M) to refine the F_5 algorithm in order to detect more reductions to zero. In practice, our approach avoids a significant number of reductions to zero. In particular, in the case r=n-2, we prove that our new algorithm avoids all reductions to zero, and we provide a corresponding complexity analysis which improves upon the previously known estimates.
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