Geodesic
On the multiplicity of solutions of a~system of algebraic equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 239-254.

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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. 
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A. V. Pukhlikov. On the multiplicity of solutions of a~system of algebraic equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 239-254. http://geodesic.mathdoc.fr/item/TM_2012_276_a19/

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