# Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

Consider a matrix 𝐅∈𝕂[x]^m × n of univariate polynomials over a field 𝕂. We study the problem of computing the column rank profile of 𝐅. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of 𝐅 with a rank-sensitive complexity of O(r^ω-2 n (m+D)) operations in 𝕂. Here, D is the sum of row degrees of 𝐅, ω is the exponent of matrix multiplication, and O(·) hides logarithmic factors.

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01/08/2016

### Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition

The row (resp. column) rank profile of a matrix describes the stair-case...
02/01/2016

### 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 a...
02/05/2016

### A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials F∈K[x]^n × ...
09/11/2019

### Elimination-based certificates for triangular equivalence and rank profiles

In this paper, we give novel certificates for triangular equivalence and...
02/13/2017

### Certificates for triangular equivalence and rank profiles

In this paper, we give novel certificates for triangular equivalence and...
07/14/2016

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

Given a nonsingular n × n matrix of univariate polynomials over a field ...
10/09/2020

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

This paper describes an algorithm which computes the characteristic poly...