arXiv subjects
Publications and source records attributed to David C. Thompson.
Sturm's Theorem is a fundamental 19th century result relating the number of real roots of a polynomial $f$ in an interval to the number of sign alternations in a sequence of polynomial division-like calculations. We provide a short direct proof of Sturm's Theorem, including the numerically vexing case (ignored in many published accounts) where an interval endpoint is a root of $f$.
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 quadratic in the sparse encoding. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. Along the way, we extend the theory of A-discriminants to real exponents and certain exponential sums, and find new and natural NP_R-complete problems.