Abstract
Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$${a}_{j,i}\in {F}_{p}$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$${a}_{j,1}+\cdots +{a}_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m\times m$minor of the$$m\times k$$$m\times k$matrix$$(a_{j,i})_{j,i}$$${\left({a}_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq {F}_{p}^{n}$of size$$|A|> C\cdot \Gamma ^n$$$\left|A\right|>C\xb7{\Gamma}^{n}$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$${x}_{1},\cdots ,{x}_{k}\in A$are all distinct. Here,Cand$$\Gamma $$$\Gamma $are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$

. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$in the solution$$(x_1,\dots ,x_k)\in A^k$$$({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{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.