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On an Erdős-type conjecture on $\mathbb{F}_q[x]$

Rongyin Wang

TL;DR

This work extends Erdős' congruence-covering phenomenon from integers to the polynomial ring \(\mathbb{F}_q[x]\), proving that any collection of \(n\) cosets of ideals that covers all polynomials of degree less than \(n\) must cover the entire ring, with a strong result established first for \(\mathbb{F}_2[x]\) and then generalized to arbitrary \(\mathbb{F}_q[x]\). The authors adapt the Crittenden–Vanden Eynden approach, employing inclusion–exclusion, a refinement via irreducible factors, and an adjustment method to derive lower bounds on uncovered polynomials and to preclude counterexamples for large \(n\); small-\(n\) cases are checked directly. A key contribution is the sharpness result in \(\mathbb{F}_2[x]\) and a clear path to the general \(\mathbb{F}_q[x]\) setting, together with a conjectured improvement of the degree bound to \(\deg(g_0)<\tfrac{n}{q-1}\). The work deepens the Erdős-type covering theory in function fields and introduces techniques that leverage irreducible polynomial counts and CRT-based decompositions in polynomial rings.

Abstract

P. Erdős conjectured in 1962 that on the ring $\mathbb{Z}$, every set of $n$ congruence classes in $\mathbb{Z}$ that covers the first $2^n$ positive integers also covers the ring $\mathbb{Z}$. This conjecture was first confirmed in 1970 by R. B. Crittenden and C. L. Vanden Eynden. Later, in 2019, P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba provided a more transparent proof. In this paper, we follow the approach used by R. B. Crittenden and C. L. Vanden Eynden to prove the generalized Erdős' conjecture in the setting of polynomial rings over finite fields. We prove that every set of $n$ cosets of ideals in $\mathbb F_q[x]$ that covers all polynomials whose degree is less than $n$ covers the ring $\mathbb{F}_q[x]$.

On an Erdős-type conjecture on $\mathbb{F}_q[x]$

TL;DR

This work extends Erdős' congruence-covering phenomenon from integers to the polynomial ring , proving that any collection of cosets of ideals that covers all polynomials of degree less than must cover the entire ring, with a strong result established first for and then generalized to arbitrary . The authors adapt the Crittenden–Vanden Eynden approach, employing inclusion–exclusion, a refinement via irreducible factors, and an adjustment method to derive lower bounds on uncovered polynomials and to preclude counterexamples for large ; small- cases are checked directly. A key contribution is the sharpness result in and a clear path to the general setting, together with a conjectured improvement of the degree bound to \(\deg(g_0)<\tfrac{n}{q-1}\). The work deepens the Erdős-type covering theory in function fields and introduces techniques that leverage irreducible polynomial counts and CRT-based decompositions in polynomial rings.

Abstract

P. Erdős conjectured in 1962 that on the ring , every set of congruence classes in that covers the first positive integers also covers the ring . This conjecture was first confirmed in 1970 by R. B. Crittenden and C. L. Vanden Eynden. Later, in 2019, P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba provided a more transparent proof. In this paper, we follow the approach used by R. B. Crittenden and C. L. Vanden Eynden to prove the generalized Erdős' conjecture in the setting of polynomial rings over finite fields. We prove that every set of cosets of ideals in that covers all polynomials whose degree is less than covers the ring .
Paper Structure (6 sections, 8 theorems, 36 equations)

This paper contains 6 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Observations for $\mathbb{F}_2[x]$
  3. Two Lemmas
  4. Proof of Proposition \ref{['keyprop']}
  5. Proof of Theorem 1
  6. Proof of Theoerm \ref{['qthm']} and some discussion

Key Result

Theorem 1

Let $f_1(x), f_2(x), \cdots, f_n(x), r_1(x), r_2(x), \cdots, r_n(x)$ be polynomials in $\mathbb{F}_2[x]$ satisfying $\deg(r_i(x)) < \deg(f_i(x))$ for $1\leq i\leq n$. Suppose that there exists some polynomial $g(x) \in \mathbb{F}_2[x]$ that satisfies none of the congruence equations for all $1 \leq i \leq n$, then there exists a polynomial $g_0(x) \in \mathbb{F}_2[x]$ that satisfies none of the c

Theorems & Definitions (15)

  • Definition
  • Theorem 1
  • Theorem 2
  • Conjecture
  • Proposition 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 5 more