Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

@article{Labahn2017FastDC,
title={Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix},
author={George Labahn and Vincent Neiger and Wei Zhou},
journal={J. Complex.},
year={2017},
volume={42},
pages={44-71}
}
• Published 14 July 2016
• Computer Science, Mathematics
• J. Complex.
22 Citations

On the complexity of computing integral bases of function fields

The problem of computing an integral basis of the algebraic function field K(\mathcal{C}) and new complexity bounds for three known algorithms dealing with this problem are given.

On Computing the Resultant of Generic Bivariate Polynomials

An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K using (n2 - 1/ømega d) 1+o(1) arithmetic operations in K, where two n x n matrices are multiplied using O(nømega) operations.

Computing Hermite Normal Form Faster via Solving System of Linear Equations

• Computer Science, Mathematics
ISSAC
• 2019
A new technique to compute the HNF for integer matrices via solving a system of linear equations, with which the authors can control the intermediate numbers more tightly is proposed.

High-order lifting for polynomial Sylvester matrices

• Mathematics
• 2022
A new algorithm is presented for computing the resultant of two “su ﬃ ciently generic” bivariate polynomials over an arbitrary ﬁeld. For such p and q in K [ x , y ] of degree d in x and n in y , the

Computing Canonical Bases of Modules of Univariate Relations

• Computer Science, Mathematics
ISSAC
• 2017
The triangular shape of M is exploited to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms and relies on high-order lifting to perform fast modular products of polynomial matrices of the form P F mod M.

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.

Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices

• Mathematics, Computer Science
ISSAC
• 2021
A first baby steps/giant steps approach --directly derived using known techniques on structured matrices-- gives a randomized Monte Carlo algorithm for the minimal polynomial of an (n x n) Toeplitz or Hankel-like matrix of displacement rank α using arithmetic operations.

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

• Mathematics, Computer Science
ISSAC
• 2018
Deterministic, fast algorithms for rectangular input matrices for normal forms for matrices over the univariate polynomials are presented.

The shortest even cycle problem is tractable

• Computer Science, Mathematics
STOC
• 2022
A family of finite commutative rings of characteristic 4 are designed that simultaneously give a nondegenerate representation for the generating polynomial identity via the permanent and the determinant, support efficient permanent computations by extension of Valiant’s techniques, and enable emulation of finite-field arithmetic in characteristic 2.

References

SHOWING 1-10 OF 44 REFERENCES

On the complexity of computing determinants

• Mathematics, Computer Science
computational complexity
• 2004
New baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems that deterministically compute the determinant, characteristic polynomial and adjoint of A with n3.2+o(1) and O(n2.697263) ring additions, subtractions and multiplications are presented.

Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix

The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499–507].

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.

On the complexity of inverting integer and polynomial matrices

An algorithm is presented that probabilistically computes the exact inverse of a nonsingular n × n integer matrix A using $${({n^3(\log||A||+\log \kappa(A)))}^{1+o(1)}}$$(n3(log||A||+logκ(A)))1+o(1)

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

On computing the determinant and Smith form of an integer matrix

• Computer Science
Proceedings 41st Annual Symposium on Foundations of Computer Science
• 2000
A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix by computing the Smith form of the integer matrix an extremely useful canonical form in itself.

Modular Computation for Matrices of Ore Polynomials

• Computer Science, Mathematics
• 2007
The algorithms improve on existing fraction-free algorithms and can be viewed as generalizations of the work of Li and Nemes on GCRDs and LCLMs of Ore polynomials and determine the appropriate normalization, as well as bound the number of homomorphic images required.

Computing Popov and Hermite forms of polynomial matrices

These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials to the Hermite normal form.

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.