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 $f_1=...= f_M =0$ 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 $\sqrt{M}\exp[ω\sqrt{M}]$, where $ω>0$ is a certain constant.

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