Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

Abstract

AbstractLet us fix a prime p and a homogeneous system of m linear equations $$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ a j , 1 x 1 + ⋯ + a j , k x k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m with coefficients $$a_{j,i}\in \mathbb {F}_p$$ a j , i ∈ F p . Suppose that $$k\ge 3m$$ k ≥ 3 m , that $$a_{j,1}+\dots +a_{j,k}=0$$ a j , 1 + ⋯ + a j , k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m and that every $$m\times m$$ m × m minor of the $$m\times k$$ m × k matrix $$(a_{j,i})_{j,i}$$ ( a j , i ) j , i is non-singular. Then we prove that for any (large) n, any subset $$A\subseteq \mathbb {F}_p^n$$ A ⊆ F p n of size $$|A|> C\cdot \Gamma ^n$$ | A | > C · Γ n contains a solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to the given system of equations such that the vectors $$x_1,\dots ,x_k\in A$$ x 1 , ⋯ , x k ∈ A are all distinct. Here, C and $$\Gamma $$ Γ are constants only depending on p, m and k such that $$\Gamma <p$$ Γ < p . The crucial point here is the condition for the vectors $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k in the solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to be distinct. If we relax this condition and only demand that $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.