On the enumeration of polynomials with prescribed factorization pattern
Simon Kuttner, Qiang Wang

TL;DR
This paper develops generating function techniques over group rings to precisely count finite field polynomials with specific initial coefficients and factorization patterns, including smoothness and trace constraints.
Contribution
It introduces new exact formulas for counting polynomials with prescribed coefficients and factorization patterns over finite fields, advancing enumeration methods.
Findings
Derived formulas for monic n-smooth polynomials of degree m
Counted polynomials with prescribed trace coefficient
Provided enumeration techniques using generating functions
Abstract
We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic -smooth polynomial of degree over a finite field, as well as the number of monic -smooth polynomial of degree with the prescribed trace coefficient.
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Taxonomy
TopicsCoding theory and cryptography · Cellular Automata and Applications · semigroups and automata theory
