Randomization, Sums of Squares, and Faster Real Root Counting for Tetranomials and Beyond
CoRR(2011)
摘要
Suppose f is a real univariate polynomial of degree D with exactly 4 monomial
terms. We present an algorithm, with complexity polynomial in log D on average
(relative to the stable log-uniform measure), for counting the number of real
roots of f. The best previous algorithms had complexity super-linear in D. We
also discuss connections to sums of squares and A-discriminants, including
explicit obstructions to expressing positive definite sparse polynomials as
sums of squares of few sparse polynomials. Our key tool is the introduction of
efficiently computable chamber cones, bounding regions in coefficient space
where the number of real roots of f can be computed easily. Much of our theory
extends to n-variate (n+3)-nomials.
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关键词
sum of squares,algebraic geometry,positive definite
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