Computing local intersection multiplicity of plane curves via blowup

TL;DR

This paper establishes an equivalence between intersection multiplicity of plane curves, as defined by Fulton, and the multiplicity obtained through blowup, providing an algorithm with complexity estimates for polynomials over various fields.

Contribution

It introduces a new algorithm for computing intersection multiplicity via blowup and analyzes its complexity, extending to polynomials over algebraic extensions.

Findings

Proves equivalence of intersection multiplicity definitions

Provides an algorithm for multiplicity computation

Estimates the algorithm's complexity

Abstract

We prove that intersection multiplicity of two plane curves defined by Fulton's axioms is equivalent to the multiplicity computed using blowup. The algorithm based on the latter is presented and its complexity is estimated. We compute for polynomials over $\Q$ and its algebraic extensions.