On the multiplicity of solutions of a system of algebraic equations

We obtain upper bounds for the multiplicity of an isolated solution of a system of equations $f_1=\dots=f_M=0$ in $M$ variables, where the set of polynomials $(f_1,\dots,f_M)$ 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 as $M\to\infty$ this multiplicity grows no faster than $\sqrt M\exp[\omega\sqrt M]$, where $\omega>0$ is a certain constant.