Computing and using minimal polynomials. (September 2020) - Abbott, John; Bigatti, Anna Maria; Palezzato, Elisa; Robbiano, Lorenzo
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.