Using Smith Normal Forms and mu-Bases to Compute all the Singularities of Rational Planar Curves

TL;DR

This paper proves a conjecture on calculating all singularities, including infinitely near ones, of rational planar curves using Smith normal forms of Bezout resultant matrices derived from mu-bases.

Contribution

It introduces a method to determine all singularities of rational planar curves via Smith normal forms, confirming a conjecture by Chen, Wang, and Liu.

Findings

Validated the conjecture on singularity computation

Provided a systematic approach using Smith normal forms

Enhanced understanding of singularities in rational curves

Abstract

We prove the conjecture of Chen, Wang and Liu in [8] concerning how to calculate the parameter values corresponding to all the singu- larities, including the infinitely near singularities, of rational planar curves from the Smith normal forms of certain Bezout resultant ma- trices derived from mu-bases.