Descartes' rule for trinomials in the plane and beyond
Abstract: Descartes' Rule (discovered no later than 1641) states that a univariate polynomial with exactly m monomial terms has at most m-1 positive roots, and the signs of the coefficients more closely determine the number of positive roots. Strangely, more than three and a half centuries later, there is no analogous sharp bound for systems of multivariate polynomials.
A recent example of Bertrand Haas shows that a pair of real bivariate polynomial equations (where each equation has at most 3 monomial terms) can have as many as 5 roots in the positive quadrant. However, until recently, the best upper bound on the number of roots independent of the degrees (following from more general results of Khovanski) was 248,832. We give an elementary proof that 5 is the correct maximum. Our proof also generalizes to real exponents, systems where one of the equations has more than 3 terms, and counting connected components. Some of the results presented are joint work with Tien-Yien Li and Xiaoshen Wang.