Irreducibility algorithm for the Weierstrass polynomials of two complex variables and the Puiseux expansions: Part[A], Part[B], Part[C]

TL;DR

This paper develops an explicit, elementary algorithm for determining irreducibility of Weierstrass polynomials in two complex variables and computing Puiseux expansions, advancing algebraic geometry methods.

Contribution

It presents a complete, rigorously computable algorithm for irreducibility and Puiseux expansions of W-polys, structured across three detailed parts.

Findings

Algorithm explicitly computes irreducibility of W-polys.

Method rigorously derives Puiseux expansions.

Provides a comprehensive approach in algebraic geometry.

Abstract

It is very fundamental to study irreducible plane curve singularities in algebraic geometry. The contents of the paper consist of three parts, called Part[A], Part[B] and Part[C] with Good Appendix. Our aim is to prove by Part[B] and Part[C] that a complete irreducibility algorithm for the Weierstrass polynomial of two complex variables and the Puiseux expansions in Part[A] can be explicitly and rigorously computable in an elementary way, as follows. For brevity, Weierstrass polynomials may be written by W-polys throughout this paper.