Faster real feasibility via circuit discriminants

Authors:
Frédéric Bihan;J. Maurice Rojas;Casey E. Stella
Affiliations:
Universite de Savoie, Le Bourget-du-Lac, France;Texas A&M University, College Station, TX, USA;ION, Shaker Heights, OH, USA
Venue:
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Year:
2009

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Abstract

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 nn. 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.