| Summary: | <p>Sárközy proved that dense sets of integers contain two elements differing by a kth power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy’s theorem for polynomials over Fq with polynomial dependencies in the parameters</p> <br/> <p>More precisely, let Pq,n be the space of polynomials over Fq of degree < n in an indeterminate T. Let k ⩾ 2 be an integer and let q be a prime power. Set c(k, q) := (2k^2 Dq (k)^2 log q)^−1, where Dq (k) is the sum of the digits of k in base q. If A ⊂ Pq,n is a set with |A| > 2q^(1−c(k,q))n, then A contains distinct polynomials p(T ), p′(T ) such that p(T ) − p′(T ) = b(T)^k for some b ∈ Fq [T ].</p>
|
|---|