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Splitting sums of binary polynomials

Luis H. Gallardo

Abstract

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of the form $x^k(x+1)^{\ell}$.

Splitting sums of binary polynomials

Abstract

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that is the minimal number such that the sums of any two distinct polynomials in a set of polynomials over cannot all be of the form .
Paper Structure (6 sections, 7 theorems, 36 equations)

This paper contains 6 sections, 7 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Tools
  3. Proof of Theorem \ref{['Nsedaka']}
  4. Proof of Theorem \ref{['Nsedaka1']}
  5. Conclusion
  6. Acknowledgments

Key Result

Theorem 1

Assume that $a, b, c \in \mathbb{F}_2[x]$ satisfy That is, with $(a_j, b_j) \neq (0, 0)$, and $a_1 \leq a_2 \leq a_3$. Then (up to switching $x$ and $x+1$), the following holds: for some non-negative integer $s$;

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 2 more