Optimization and NP_R-completeness of certain fewnomials

Authors:
Philippe P. Pébay;J. Maurice Rojas;David C. Thompson
Affiliations:
Sandia National Laborotories, Livermore, CA, USA;Texas A&M University, College Station, TX, USA;Sandia National Laboratories, Livermore, CA, USA
Venue:
Proceedings of the 2009 conference on Symbolic numeric computation
Year:
2009

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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 polynomial in n and the logarithm of a certain condition number. For the special case of polynomials (i.e., integer exponents), the log of our condition number is sub-quadratic in the sparse size. The best previous complexity bounds were exponential in the size, even for n fixed. Along the way, we partially extend the theory of A-discriminants to real exponents and exponential sums, and find new and natural NPR-complete problems.