Computing minimal nullspace bases

Abstract

In this paper we present a deterministic algorithm for the computation of a minimal nullspace basis of an m x n input matrix of univariate polynomials over a field K with m ≤ n. This algorithm computes a minimal nullspace basis of a degree d input matrix with a cost of O~ (nω ⌈md/n⌉) field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. The same algorithm also works in the more general situation on computing a shifted minimal nullspace basis, with a given degree shift [equation] whose entries bound the corresponding column degrees of the input matrix. In this case if ρ is the sum of the m largest entries of s, then a s-minimal right nullspace basis can be computed with a cost of O~(nωρ/m) field operations.