Faster real feasibility via circuit discriminants

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 −→ ∞. 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. Keywords sparse, real, feasibility, polynomial-time, discriminant chamber, linear forms in logarithms

• Type: Conference
• Year: 2009
• Where: ISSAC
• Authors: Frédéric Bihan, J. Maurice Rojas, Casey E. Stella
• Keywords: ISSAC 2009, Linear Forms, Mathematics, Real Roots, Sparse Encoding