多项式矩阵的秩和小零空间基的计算
Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
摘要：
我们把一元多项式矩阵的秩和零空间基的计算问题归结为多项式矩阵乘法问题。对于域K上的一个n × n阶d矩阵，我们给出了一个秩和零空间算法，其运算次数与两个n维和d阶矩阵相乘的运算次数大致相同。如果后面的乘法在MM（n，d）=softO（n^omega d）操作中完成，其中omega是K上矩阵乘法的指数，则该算法在K中使用softO（MM（n，d））操作。softO表示法表示一些缺失的对数因子。该方法是随机的拉斯维加斯认证。我们部分地通过矩阵Hensel高阶提升和矩阵最小分式重构的结合，以及通过在被视为K[x]-模的零空间中计算最小或小度向量来实现我们的结果
英文摘要：
We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module
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