On the multiplicity of solutions of a system of algebraic equations

TL;DR

This paper establishes upper bounds on the multiplicity of isolated solutions for systems of algebraic equations, showing it grows at most exponentially with the square root of the number of variables.

Contribution

It provides new asymptotic bounds for solution multiplicities in polynomial systems in general position, extending understanding of solution behavior as variables increase.

Findings

Multiplicity grows no faster than b0b4b0b4b0b4[b0b4b0b4b0b4b0b4b0b4b0b4b0b4b0b4b0b4],

Solution multiplicity is bounded by an exponential function of b0b4b0b4b0b4b0b4b0b4b0b4b0b4b0b4b0b4 in terms of the number of variables.

The bounds are valid for systems in general position within a subvariety of bounded codimension.

Abstract

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 $\sqrt{M}\exp[\omega\sqrt{M}]$, where $\omega>0$ is a certain constant.