Uniform Probability and Natural Density of Mutually Left Coprime\n Polynomial Matrices over Finite Fields

2017/02/27 by Julia Lieb, Lieb, Julia · 1 citation
Computer Science · Engineering · #Cellular Automata and Applications #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1702.08312

openalex publication_date 2017/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the uniform probability that finitely many polynomials over a\nfinite field are pairwise coprime and compare the result with the formula one\ngets using the natural density as probability measure. It will turn out that\nthe formulas for the two considered probability measures asymptotically\ncoincide but differ in the exact values. Moreover, we calculate the natural\ndensity of mutually left coprime polynomial matrices and compare the result\nwith the formula one gets using the uniform probability distribution. The\nachieved estimations are not as precise as in the scalar case but again we can\nshow asymptotic coincidence.\n

Citations

• Connections between Linear Systems and Convolutional Codes

Cited by

• Counting w-coprime S-integers and S-integral ideals in positive characteristic

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