New Sparse Multivariate Polynomial Factorization Algorithms over Integers

We propose two algorithms for sparse polynomial factorization over integers. The first one has good practical performance and is efficient for factoring polynomials with sparse irreducible factors. The second one is based on the effective Hilbert irreducibility theorem and has complexity polynomial in the sizes of the input and output, and the partial degree. At high level, the algorithms follow the standard approaches by reducing multi-variate polynomial factorization to univariate or bivariate polynomial factorization. Our main contributions are twofold. First, a new variable substitution is given, which reduces the multi-variate polynomial to a separated one, that is, the coefficients of its factors in a main variable are monomials. Second, "good" primes are selected such that the multi-variate factors can be recovered from the univariate or bivariate factors by direct division of the primes. As a consequence, the multivariate Hensel lifting in previous methods is avoided.