# Implementations of efficient univariate polynomial matrix algorithms and application to bivariate resultants

## TL;DR

This paper explores the practical implementation of efficient univariate polynomial matrix algorithms, demonstrating improved performance in computing bivariate resultants, with implications for error-correcting codes and polynomial systems.

## Contribution

It provides the first practical implementation of several fundamental polynomial matrix operations and applies these to enhance bivariate resultant computations.

## Key findings

- Improved performance over existing methods for large parameters
- Successful implementation of key polynomial matrix operations
- Enhanced algorithms for bivariate resultants

## Abstract

Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of the practical impact of these algorithms, e.g. on applications such as decoding some error-correcting codes and solving polynomial systems or structured linear systems. In this paper, we discuss implementation aspects for most fundamental operations: multiplication, truncated inversion, approximants, interpolants, kernels, linear system solving, determinant, and basis reduction. We focus on prime fields with a word-size modulus, relying on Shoup's C++ library NTL. Combining these new tools to implement variants of Villard's algorithm for the resultant of generic bivariate polynomials (ISSAC 2018), we get better performance than the state of the art for large parameters.