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Romanoff type theorems for polynomials over finite fields
Abstract
Given a polynomial g of degree δ>0 over a finite field, we study monic polynomials f of degree n that can be written as f=h+gk, where h is a monic irreducible polynomial of degree n and k∈ℕ with δk<n. Following Erdős, we show that some f admit at least clog n such representations. If δρg<1, where ρg=φq(g)/|g|, then a positive proportion of degree-n polynomials admits at least two such representations. More generally, for sums of powers g⌊ kiri⌋, we prove that a positive proportion of polynomials of degree n is representable whenever ∑i=1tri−1≥ 1, while certain residue classes modulo some irreducible polynomial contain no representable polynomial.
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BibTeX
@misc{wang2026romanoff,
title = {{Romanoff type theorems for polynomials over finite fields}},
author = {Enci Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22875},
eprint = {2609.22875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22875},
}
RIS
TY - UNPB
TI - Romanoff type theorems for polynomials over finite fields
AU - Enci Wang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22875
UR - https://arxiv.org/abs/2609.22875
AB - Given a polynomial g of degree δ>0 over a finite field, we study monic polynomials f of degree n that can be written as f=h+g^k, where h is a monic irreducible polynomial of degree n and k∈ℕ with δk<n. Following Erdős, we show that some f admit at least clog n such representations. If δρ_g<1, where ρ_g=φ_q(g)/|g|, then a positive proportion of degree-n polynomials admits at least two such representations. More generally, for sums of powers g^⌊ k_i^r_i⌋, we prove that a positive proportion of polynomials of degree n is representable whenever ∑_i=1^tr_i^−1≥ 1, while certain residue classes modulo some irreducible polynomial contain no representable polynomial.
ER -