# 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} }

## 22 Citations

### On the complexity of computing integral bases of function fields

- Mathematics, Computer ScienceCASC
- 2020

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

- Mathematics, Computer ScienceISSAC
- 2018

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, MathematicsISSAC
- 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.

### Deterministic computation of the characteristic polynomial in the time of matrix multiplication

- Mathematics, Computer ScienceJ. Complex.
- 2021

### 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, MathematicsISSAC
- 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 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.

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

- Mathematics, Computer ScienceISSAC
- 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 ScienceISSAC
- 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, MathematicsSTOC
- 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.

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