Doubly-Exponential Growth of the Number of Vectors of Multipilicities for Solutions of Systems of Polynomial

Abstract

Earlier, D. Yu. Grigoriev and N. N. Vorob'yov obtained an upper bound $d^{O(\left( {_n^{n + d} } \right))}$ for the number of vectors of multiplicities for solutions of systems of the form $g1 = \cdots = gn = 0$ (assuming that the system has a finite number of solutions). In the system above, $g1, \ldots ,gn \in F[X1, \ldots ,Xn]$ are polynomials of degrees $\deg (gi) \leqslant d$. In the present paper, we show that the order of growth of this bound is close to the exact one. In particular, in the case d=n, the construction provides a doubly-exponential (in n) number of vectors of multiplicities. Bibliography: 4 titles.