Real Factorization of Multivariate Polynomials with Integer Coefficients

Abstract

In the recent papers, a new efficient probabilistic semi-numerical absolute (i.e., complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It is based on a simple property of the monomials arising after a generic linear change of coordinates for bivariate polynomials and on a deep result of complex algebraic geometry. Here, we consider the a priori simpler problem of factorization over the field of reals. We briefly review our algorithm for complex factorization and adapt it to solving the problem on the field of reals. This allows us to spare a significant part of the computations and to improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of \(\mathbb{Q}\). Bibliography: 15 titles.