Randomization , Sums of Squares , Near-Circuits , and Faster Real Root Counting

Suppose thatf is a real univariate polynomial of degree D with exactly 4 monomial terms. We present a deterministic algorithm of complexi ty polynomial in logD that, for most inputs, counts the number of real roots of f . The best previous algorithms have complexity super-linear in D. We also discuss connections to sums of squares and Adiscriminants, including explicit obstructions to express ing positive definite sparse polynomials as sums of squares of few sparse polynomials. Our key the oretical tool is the introduction of efficiently computable chamber cones , which bound regions in coefficient space where the number of real roots of f can be computed easily. Much of our theory extends ton-variate(n+3)-nomials.