# Computing minimal nullspace bases

@inproceedings{Zhou2012ComputingMN, title={Computing minimal nullspace bases}, author={Wei Zhou and George Labahn and Arne Storjohann}, booktitle={ISSAC}, year={2012} }

In this paper we present a deterministic algorithm for the computation of a minimal nullspace basis of an <i>m</i> x <i>n</i> input matrix of univariate polynomials over a field K with <i>m</i> ≤ <i>n</i>. This algorithm computes a minimal nullspace basis of a degree <i>d</i> input matrix with a cost of <i>O</i>~ (<i>n</i><sub>ω</sub> ⌈<i>md</i>/<i>n</i>⌉) field operations in K. Here the soft-<i>O</i> notation is Big-<i>O</i> with log factors removed while ω is the exponent of matrix…

## 29 Citations

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To obtain the target cost for any shift, the properties of the output bases are strengthened, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always O(m σ), and a divide-and-conquer scheme is designed.

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- MathematicsArXiv
- 2016

The method relies of a fast algorithm for determining the diagonal entries of its Hermite normal form, having as cost $O^{\sim}\left(n^{\omega}s\right)$ operations with $s$ the average of the column degrees of $\mathbf{F}$.

Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions

- Computer ScienceISSAC
- 2022

The Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case, which improves upon both state-of-the-art complexity bounds O~(tD2) and O ~(Dω, since ω<3 and t≤D), and confirms the high practical benefit.

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