On a Conjecture About the Sum-Freedom of the Binary Multiplicative Inverse Function

TL;DR

The paper proves Carlet's Conjecture for composite $n$ by recasting the problem as the existence of zero-sum $\mathbb{F}_2$-subspaces of $\mathbb{F}_{2^n}$ and linking this to zeros of a family of polynomials $F_{k,l}$ derived from determinant polynomials $\Delta$ and $\Delta_1$. It establishes absolute irreducibility of $F_{k,l}$ and leverages the Hasse-Weil bound to guarantee nontrivial zeros, enabling the construction and extension of zero-sum subspaces across dimensions. A lifting argument guided by the minimal prime divisor $l$ of $n$, together with propagation via auxiliary theorems, shows that all $k$ with $3\le k\le n-3$ are realized as sum-free, for $n$ not prime. This integrates algebraic and geometric tools with prior partial results (e.g., when $n$ is even, divisible by $3$ or $5$) to resolve the conjecture in the composite-$n$ case and highlights methods that may advance the prime-$n$ case. The work advances understanding of sum-free properties of the binary multiplicative inverse and illustrates a unifying approach that could extend to related families of functions.

Abstract

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional $\Bbb F_2$-subspace $E$ of $\Bbb F_{2^n}$ such that $\sum_{0\ne\in E}1/u=0$. We confirm this conjecture when $n$ is not a prime.

Theorems & Definitions (15)

• Conjecture 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof