Abstract
A non-zero polynomial of degree d has at most d complex roots. Moreover, Descartes proved that the number of real roots can be also bounded by a function on the number of monomials. A polynomial with t monomials has at most 2t-1 real roots.
What happens if we consider the solutions of a system of polynomials? In the complex case, Bézout's Theorem bounds the number of solutions by the product of the degrees. Is there, in the real case, a similar bound which only depends on the number of monomials? Khovanskií proved that this bound exists, but the given bound is exponential. Is it the optimal bound? This is an open question.
A particular case, known as Sevostyanov's problem, is the case of a system of one degree d polynomial and of one t-sparse polynomial. We will present in this talk a polynomial bound with respect to t and d for this problem. It is a joint work with Pascal Koiran and Natacha Portier.