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 as M → ∞ this multiplicity grows no faster than \(\sqrt M \exp \left {\omega \sqrt M } \right]\), where ω > 0 is a certain constant.