Special-case Algorithms for Blackbox Radical Membership, Nullstellensatz and Transcendence Degree
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Radical membership testing, and the special case of Hilbert's Nullstellensatz (HN), is a fundamental computational algebra problem. It is NP-hard; and has a famous PSPACE algorithm due to effective Nullstellensatz bounds. We identify a useful case of these problems where practical algorithms, and improved bounds, could be given, when the transcendence degree r of the input polynomials is smaller than the number of variables n. If d is the degree bound on the input polynomials, then we solve radical membership (even if input polynomials are blackboxes) in around d^r time. The prior best was > d^n time (always, d^n≥ d^r). Also, we significantly improve effective Nullstellensatz degree-bound, when r≪ n. Structurally, our proof shows that these problems reduce to the case of r+1 polynomials of transcendence degree ≥ r. This input instance (corresponding to none or a unique annihilator) is at the core of HN's hardness. Our proof methods invoke basic algebraic-geometry.
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