Extended Hardness Results for Approximate Gröbner Basis Computation

Two models were recently proposed to explore the robust hardness of Gröbner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gröbner basis for the ideal generated by the remaining polynomials. For the q-Fractional Gröbner Basis Problem the algorithm is allowed to ignore a constant (1 − q)-fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a (3/10 − ǫ)-fraction of the polynomials to ignore, and need only compute a Gröbner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless P = NP ). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum degree 2: for the q-Fractional model a (1/5−ǫ) fraction of the polynomials may be ignored without losing provable NP-Hardness. Both theorems hold even if every polynomial contains at most three distinct variables. Finally, for the Strong cpartial Gröbner Basis Problem of De Loera et al. we give conditional results that depend on famous (unresolved) conjectures of Khot and Dinur, et al.