A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over [formula omitted

Abstract

In this paper, we give the first polynomial time algorithm to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Gröbner basis of the Z[x]-module generated by the column vectors of F. The algorithm has polynomial bit size computational complexities and is also shown to be practically more efficient than existing algorithms. The algorithm is based on three key ingredients. First, an F4 style algorithm to compute the Gröbner basis is adopted, where a novel prolongation is designed such that the sizes of coefficient matrices under consideration are nicely controlled. Second, the complexity bound of the algorithm is achieved by a nice estimation for the degree and height bounds of the polynomials in the generalized Hermite normal form. Third, fast algorithms to compute Hermite normal forms of matrices over Z are used as the computational tool.