# Computing and Using Minimal Polynomials

## TL;DR

This paper introduces efficient algorithms for computing minimal polynomials of elements in zero-dimensional ideals, with modular methods for rational coefficients, and explores applications like radical and primary decomposition, and ideal property testing.

## Contribution

It presents new algorithms for minimal polynomial computation, including modular techniques for rational coefficients, and demonstrates their applications in ideal decomposition and property testing.

## Key findings

- Algorithms outperform existing methods in efficiency
- Modular techniques guarantee correct results over Q
- Applications include radical and primary decomposition, and ideal property testing

## Abstract

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality.