Equidistribution of polynomial sequences in function fields: resolution of a conjecture

Let Fq be the finite field of q elements having characteristic p, and denote by K_∞= Fq((1/t)) the field of formal Laurent series in $1/t$. We consider the equidistribution in T= K_∞/ Fq[t] of the values of polynomials f(u)∈ K_∞ [u] as u varies over Fq[t]. Let K be a finite set of positive integers, and suppose that αᵣ∈ K_∞ for r∈ K∪ \0\. We show that the polynomial ∑_r∈ K∪\0\αᵣuʳ is equidistributed in T whenever αₖ is irrational for some k∈ K satisfying p k, and also pᵛk∈ K for any positive integer v. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.