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

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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.
Philippe P. Pébay, J. Maurice Rojas, David
Added 15 May 2011
Updated 15 May 2011
Type Journal
Year 2011
Where TCS
Authors Philippe P. Pébay, J. Maurice Rojas, David C. Thompson
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