Bivariate factorization using a critical fiber

TL;DR

This paper extends bivariate polynomial factorization techniques to critical fibers, offering new strategies that improve theoretical complexity especially for non-degenerate polynomials with respect to P-adic Newton polytopes.

Contribution

It generalizes the lifting and recombination scheme for critical fibers, introducing strategies tailored to ramification complexity and analyzing benefits for non-degenerate polynomials.

Findings

Working along a critical fiber reduces the number of analytic factors to recombine.

The approach achieves better theoretical complexity in certain cases.

Special attention is given to non-degenerate polynomials with respect to P-adic Newton polytopes.

Abstract

We generalize the classical lifting and recombination scheme for rational and absolute factorization of bivariate polynomials to the case of a critical fiber. We explore different strategies for recombinations of the analytic factors, depending on the complexity of the ramification. We show that working along a critical fiber leads in some cases to a good theoretical complexity, due to the smaller number of analytic factors to recombine. We pay a particular attention to the case of polynomials that are non degenerate with respect to their P-adic Newton polytopes.