Faster Real Feasibility via Circuit Discriminants

Details Faster Real Feasibility via Circuit Discriminants Faster Real Feasibility via Circuit Discriminants

2009-01-28

arxiv.org/abs/0901.4400v2

Mathematics

Algebraic Geometry

12 pages in double column ACM format. Submitted to a conference. Significantly improves and simplifies the algorithms and comp

Scientific paper

We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a characterization of those functions k(n) such that the complexity of detecting real roots for n-variate (n+k(n))-nomials transitions from P to NP-hardness as n tends to infinity. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of A-discriminants of n-variate (n+k(n))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.

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