Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

We establish cancellation in short sums of certain special trace functions over 𝔽_q[u] below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the 𝔽_q[u] -analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q . A final application is to sums of trace functions over primes in 𝔽_q[u] .