Optimizing n-variate (n+k)-nomials for small k

Authors:
Philippe Pébay;J. Maurice Rojas;David C. Thompson
Affiliations:
Sandia National Laboratories, PO Box 969, MS 9159, Livermore, CA 94551, USA;TAMU 3368, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA;Sandia National Laboratories, PO Box 969, MS 9159, Livermore, CA 94551, USA
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are quadratic in n and the logarithm of a certain condition number. For the special case of n-variate (n+2)-nomials with integer exponents, the log of our condition number is sub-quadratic in the sparse size. The best previous complexity bounds were exponential in the sparse size, even for n fixed. Along the way, we partially extend the theory of Viro diagrams and A-discriminants to real exponents. We also show that, for any fixed @d0, deciding whether the supremum of an n-variate (n+n^@d)-nomial exceeds a given number is NP"R-complete.