计算Z[x]上矩阵广义Hermite范式的多项式时间算法
A Polynomial-time Algorithm to Compute Generalized Hermite Normal Form of Matrices over Z[x]
摘要：
本文给出了计算Z[x]上矩阵F的广义Hermite范式的一个多项式时间算法，即计算由F的列向量生成的Z[x]-模的约化Groebner基.该算法也被证明是实际上更有效的比现有的算法。该算法基于三个关键要素。首先，采用F4风格的算法来计算Groebner基，其中设计了一种新的延拓，使得所考虑的系数矩阵具有多项式大小。第二，使用快速算法计算Z上矩阵的Hermite范式。第三，通过对广义Hermite范式中多项式的次数和高度界的精确估计，保证了算法的复杂度。
英文摘要：
In this paper, a polynomial-time algorithm is given to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced 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 existing algorithms. The algorithm is based on three key ingredients. First, an F4 style algorithm to compute the Groebner basis is adopted, where a novel prolongation is designed such that the coefficient matrices under consideration have polynomial sizes. Second, fast algorithms to compute Hermite normal forms of matrices over Z are used. Third, the complexity of the algorithm are guaranteed by a nice estimation for the degree and height bounds of the polynomials in the generalized Hermite normal form.
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