Deciding properties of polynomials without factoring

The polynomial time algorithm of Lenstra, Lenstra, and Lovasz (1982) for factoring integer polynomials and variants thereof have been widely used to show that various computational problems in number theory have polynomial time solutions. Among them is the problem of factoring polynomials over algebraic number fields, which is used itself as a major subroutine for several other algorithms. Although a theoretical breakthrough, algorithms based on factorization of polynomials are notoriously slow and hard to implement, with running times ranging between O(n¹²) and O(n¹⁸) depending on which variant of the lattice basis reduction is used. Here, n is an upper bound for the maximum of the degrees and the bit-lengths of the coefficients of the polynomials involved. On the other hand, in many situations one does not need the full power of factorization, so one may ask whether there exist faster algorithms in these cases. In this paper we develop more efficient Monte Carlo algorithms to decide certain properties of roots of integer polynomials, without factoring them. Such problems arise, e.g., when solving systems of algebraic equations. Our methods applied to this situation thus give information about the solutions of such systems of equations