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

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 δ >0, deciding whether the supremum of an n-variate n + nδ -nomial exceeds a given number is NPR-complete.

• Type: Journal
• Year: 2011
• Where: TCS
• Authors: Philippe P. Pébay, J. Maurice Rojas, David C. Thompson
• Keywords: Condition Number, Real Exponents, Sparse Size, TCS 2011, Theoretical Computer Science