Counting polynomials over finite fields with prescribed leading coefficients and linear factors

Zhicheng Gao

doi:10.1016/j.ffa.2022.102052

Home/Archive/arXiv

arXiv · 2105.12845

Counting polynomials over finite fields with prescribed leading coefficients and linear factors

Zhicheng Gao

View original source ↗Read source document ↗Export metadata ↓

Abstract

We count the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This is equivalent to counting codewords in Reed-Solomon codes which are at a certain distance from a received word. We first apply the generating function approach, which is recently developed by the author and collaborators, to derive expressions for the number of monic polynomials with prescribed leading coefficients and linear factors. We then apply Li and Wan's sieve formula to simplify the expressions in some special cases. Our results extend and improve some recent results by Li and Wan, and Zhou, Wang and Wang.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zhicheng Gao. 2022-11-10. Counting polynomials over finite fields with prescribed leading coefficients and linear factors. https://doi.org/10.1016/j.ffa.2022.102052

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗

arXiv2026

Generalized Goulden-Yong duals and signed minimal factorizations

In this paper, we give two combinatorial ways to study signed exceptional sequences. First, we show the equivalence between one-way reflections and relatively projective representations. Secondly, we construct generalized Goulden-Yong duals using reverse Garside element actions and folded chord diagrams. We then give two applications of the generalized Goulden-Yong duals: constructing generalized Prüfer codes and counting signed factorizations using the matrix-tree theorem.

math.CO↗

arXiv2026

Explicit expressions for iterates of power series

We present several formulas for both the discrete and fractional iterates of an invertible power series $f$, using a new unifying approach based on umbral calculus. Known formulas are extended, and their proofs simplified, while new expressions are introduced. In particular, by employing $q$-calculus identities, we eliminate the requirement for $f'(0)$ to equal $1$ and the resulting general expressions for the iterative logarithm are obtained as well.

math.CO↗