On the robust hardness of Gröbner basis computation
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We introduce a new problem in the approximate computation of Gröbner bases that allows the algorithm to ignore a constant fraction of the generators - of the algorithm's choice - then compute a Gröbner basis for the remaining polynomial system. The set ignored is subject to one quite-natural structural constraint. For lexicographic orders, when the discarded fraction is less than (1/4-ϵ), for ϵ>0, we prove that this problem cannot be solved in polynomial time, even when the original polynomial system has maximum degree 3 and each polynomial contains at most 3 variables. Qualitatively, even for sparse systems composed of low-degree polynomials, we show that Gröbner basis computation is robustly hard: even producing a Gröbner basis for a large subset of the generators is NP-hard.
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