Computing and using minimal polynomials. (September 2020)

• Record Type: Journal Article
• Authors: Abbott, John; Bigatti, Anna Maria; Palezzato, Elisa; Robbiano, Lorenzo
• Is Part Of: Journal of symbolic computation. Volume 100(2020)
• Page Start: 137
• Page End: 163
• Publication Date: 2020-09
• Subjects: 13P25 -- 13P10 -- 13-04 -- 14Q10 -- 68W30; Minimal polynomial -- Gröbner bases -- Modular methods -- Radical -- Maximal -- Primary; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.07.022
• Languages: English
• ISSNs: 0747-7171

Abstract

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.