Finding solutions with distinct variables to systems of linear equations over \mathbbF_p

2021/05/14 by Lisa Sauermann, Sauermann, Lisa
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2105.06863

openalex publication_date 2021/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let us fix a prime p and a homogeneous system of m linear equations a_j,1x₁+…+a_j,kx_k=0 for j=1,…,m with coefficients a_j,i∈\mathbbF_p. Suppose that k≥ 3m, that a_j,1+…+a_j,k=0 for j=1,…,m and that every m× m minor of the m× k matrix (a_j,i)_j,i is non-singular. Then we prove that for any (large) n, any subset A⊆\mathbbF_pⁿ of size |A|> C⋅ Γⁿ contains a solution (x₁,…,x_k)∈ A^k to the given system of equations such that the vectors x₁,…,x_k∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ

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