# Computing Popov and Hermite forms of polynomial matrices

```@inproceedings{Villard1996ComputingPA,
title={Computing Popov and Hermite forms of polynomial matrices},
author={Gilles Villard},
booktitle={International Symposium on Symbolic and Algebraic Computation},
year={1996}
}```
• G. Villard
• Published in
International Symposium on…
1 October 1996
• Mathematics
For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(log~+’ (ml)) pamlel arithmetic steps and O(L(nd) ) processors if the same processor bound holds with time O (logX (rid)) for determining the lexicographically first maximal linearly independent subset…
44 Citations

### Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

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### Normal forms for general polynomial matrices

• Mathematics, Computer Science
J. Symb. Comput.
• 2006

### Fast Parallel Algorithms for Matrix Reduction to Normal Forms

• G. Villard
• Mathematics, Computer Science
Applicable Algebra in Engineering, Communication and Computing
• 1997
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### Computing Popov Forms of Polynomial Matrices

A Las Vegas algorithm that computes the Popov decomposition of matrices of full row rank is given and it is shown that the problem of transforming a row reduced matrix to Popov form is at least as hard as polynomial matrix multiplication.

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• Computer Science
• 2000
A simple algorithm to recover H when working over the polynomial ring Fx], F a eld is described, which requires O(n 3 d 2) eld operations when A 2 Fx] nn is nonsingular with degrees of entries bounded by d, and the matrix U is recovered in the same time.

### Algorithms for normal forms for matrices of polynomials and ore polynomials

• Mathematics, Computer Science
• 2003
A fraction-free algorithm for row reduction for matrices of Ore polynomials is obtained by formulating row reduction as a linear algebra problem and this algorithm is used as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov forms, and the Popov form of a polynomial matrix.

### Converting between the Popov and the Hermite form of matrices of differential operators using an FGLM-like algorithm

We consider matrices over a ring K [∂; σ, θ ] of Ore polynomials over a skew field K . Since the Popov and Hermite normal forms are both Gröbner bases (for term over position and position over term

### 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 hermite forms of polynomial matrices

• Computer Science
ISSAC '11
• 2011
This paper presents a new algorithm for computing the Hermite form of a polynomial matrix that is both softly linear in the degree d and softly cubic in the dimension n, and is randomized of the Las Vegas type.

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