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In this paper, a polynomial-time algorithm is given to compute thegeneralized Hermite normal form for a matrix F over Z[x], or equivalently, thereduced Groebner basis of the Z[x]-module generated by the column vectors of F.The algorithm is also shown to be practically more efficient than existingalgorithms. The algorithm is based on three key ingredients. First, an F4 stylealgorithm to compute the Groebner basis is adopted, where a novel prolongationis designed such that the coefficient matrices under consideration havepolynomial sizes. Second, fast algorithms to compute Hermite normal forms ofmatrices over Z are used. Third, the complexity of the algorithm are guaranteedby a nice estimation for the degree and height bounds of the polynomials in thegeneralized Hermite normal form.

A Polynomial-time Algorithm to Compute Generalized Hermite Normal Form of Matrices over Z[x]