# Computing minimal nullspace bases

@inproceedings{Zhou2012ComputingMN,
title={Computing minimal nullspace bases},
author={Wei Zhou and George Labahn and Arne Storjohann},
booktitle={ISSAC},
year={2012}
}
• Published in ISSAC 22 July 2012
• Computer Science
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
Computing column bases of polynomial matrices
• Computer Science
ISSAC '13
• 2013
This paper presents a deterministic algorithm for the computation of a column basis of an input matrix with m x n input matrix, and shows that the average column degree is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.
Unimodular completion of polynomial matrices
• Computer Science
ISSAC
• 2014
The algorithm computes a unimodular completion for a right cofactor of a column basis of <b>F</b>, or equivalently, compute a completion that preserves the generalized determinant.
Algorithm for computing μ-bases of univariate polynomials
• Mathematics, Computer Science
J. Symb. Comput.
• 2017
Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations
We give a Las Vegas algorithm which computes the shifted Popov form of an m x m nonsingular polynomial matrix of degree d in expected ~O(mω d) field operations, where ω is the exponent of matrix
A deterministic algorithm for inverting a polynomial matrix
• Computer Science, Mathematics
J. Complex.
• 2015
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
• Computer Science
ISSAC
• 2016
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.
Computing minimal interpolation bases
• Computer Science
J. Symb. Comput.
• 2017
A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix
• Mathematics
ArXiv
• 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 Science
ISSAC
• 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.

## References

SHOWING 1-10 OF 20 REFERENCES
Efficient computation of order bases
• Computer Science
ISSAC '09
• 2009
The algorithm extends earlier work of Storjohann, whose method can be used to find a subset of an order basis that is within a specified degree bound δ using <i>O</i>~(MM(<i>n,δ</i))) field operations for δ≥⌈ <i-m</i><sup>ω</sup>⌉/<i-n</i>, and order σ.
On the complexity of polynomial matrix computations
• Computer Science
ISSAC '03
• 2003
Under the straight-line program model, it is shown that multiplication is reducible to the problem of computing the coefficient of degree <i>d</i> of the determinant and algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication are proposed.
Computing the rank and a small nullspace basis of a polynomial matrix
• Mathematics, Computer Science
ISSAC
• 2005
A rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d, and the soft-O notation O~ indicates some missing logarithmic factors.
Efficient algorithms for order basis computation
• Mathematics, Computer Science
J. Symb. Comput.
• 2012
Normal forms for general polynomial matrices
• Mathematics, Computer Science
J. Symb. Comput.
• 2006
Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems
It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1}$, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.
Shifted normal forms of polynomial matrices
• Mathematics
ISSAC '99
• 1999
lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms and is able to c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.