Singular factors of rational plane curves

TL;DR

This paper provides a comprehensive factorization of invariant factors of resultant matrices for rational plane curves, linking them to singular points and multiplicity graphs, and confirms related conjectures.

Contribution

It offers a complete factorization of invariant factors and the D-resultant for rational plane curves based on singularity data, validating prior conjectures.

Findings

Complete factorization of invariant factors in terms of singular points

Validation of conjectures by Chen, Wang, and Liu

Factorization of the D-resultant using multiplicity data

Abstract

We give a complete factorization of the invariant factors of resultant matrices built from birational parameterizations of rational plane curves in terms of the singular points of the curve and their multiplicity graph. This allows us to prove the validity of some conjectures about these invariants stated by Chen, Wang and Liu in [J. Symbolic Comput. 43(2):92-117, 2008]. As a byproduct, we also give a complete factorization of the D-resultant for rational functions in terms of the similar data extracted from the multiplicities.