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 $${\mathbb {F}}_q[u]$$ below the Pólya–Vinogradov range, with savings approaching square-root as q grows. This is used to resolve -analog Chowla’s conjecture on Möbius polynomial sequences, and Bateman–Horn degree 2, for some values q. A final application primes .