New Complexity Bounds for Certain Real Fewnomial Zero Sets (Extended Abstract)

Rojas dedicates this paper to his friend, Professor Tien-Yien Li. Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m≥3, there are systems of the form (f, g) having exactly 2m − 1 roots in the positive quadrant. Even examples with m=4 having 7 positive roots were unknown before this paper, so we detail an explicit example of this form. We also present an O(n 11) upper bound for the number of diffeotopy types of the real zero set of an n-variate polynomial with n + 4 monomial terms. 1