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 $f_1=\mathrm{.}\mathrm{.}\mathrm{.}=f_M=0$ in $M$ variables, where the set of polynomials $(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_M)$ 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 $M\to \mathrm{\infty }$ that multiplicity grows not faster than $\sqrt{M}\exp [\omega \sqrt{M}]$, where $\omega >0$ is a certain constant.