The distribution of factorization patterns on linear families of polynomials over a finite field
Eda Cesaratto, Guillermo Matera, Mariana P\'erez

TL;DR
This paper estimates the distribution of factorization patterns in linear families of polynomials over finite fields, providing precise asymptotics and bounds, especially for sparse families, by analyzing associated algebraic varieties.
Contribution
It introduces a method to estimate the number of polynomials with given factorization patterns in linear families over finite fields, using algebraic geometry and symmetry considerations.
Findings
Asymptotic formulas for the count of polynomials with specific factorization patterns.
Explicit bounds for error terms depending on the family and pattern.
Reduction of counting problems to rational points on symmetric complete intersections.
Abstract
We obtain estimates on the number of elements on a linear family of monic polynomials of of degree having factorization pattern . We show that , where is the proportion of elements of the symmetric group of elements with cycle pattern and is the codimension of . Furthermore, if the family under consideration is "sparse", then . Our estimates hold for fields of characteristic greater than 2. We provide explicit upper bounds…
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Taxonomy
TopicsCoding theory and cryptography · Finite Group Theory Research · Advanced Algebra and Geometry
