Lisa Sauermann: Finding solutions with distinct variables to systems of equations over 𝔽_p

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 ∈ 𝔽_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 ⊆ (𝔽_p)ⁿ of size |A| greater than 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 < p. The crucial point here is the condition for the vectors x₁, . . .,x_k in the solution (x₁, . . .,x_k) ∈ A^k to be distinct. If we relax this condition and only demand that x₁, . . .,x_k are not all equal, then the statement would follow easily from Tao’s slicerank 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 slicerank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.