On the multiplicity of solutions of a system of algebraic equations

Abstract

We obtain upper bounds for the multiplicity of an isolated solution of a system of equations $f_1=...=f_M=0$ in $M$ variables, where the set of polynomials $(f_1,...,f_M)$ is a tuple of general position in a subvariety of a given codimension which does not exceed $M$, in the space of tuples of polynomials. It is proved that for $M\to \infty$ that multiplicity grows not faster than $\sqrt{M}\exp [\omega \sqrt{M}]$, where $\omega >0$ is a certain constant.

20 pages