Linear and quadratic uniformity of the M\"obius function over mathbb{F}_q[t]

We examine correlations of the M\"obius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} \mu(f)\chi(Q(f)) \end{equation} for an additive character $\chi$ over $\mathbb{F}_q$ and a polynomial $Q\in\mathbb{F}_q[x_0,\ldots,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\ldots, x_{n-1}$ of $f=\sum_{i< n}x_i t^i$. Like in the integers, it is reasonable to expect that, due to the random-like behaviour of $\mu$, such sums should exhibit considerable cancellation. In this paper we show that the above correlation is bounded by $O_\epsilon \left( q^{(-\frac{1}{4}+\epsilon)n} \right)$ for any $\epsilon >0$ if $Q$ is linear and $O \left( q^{-n^c} \right)$ for some absolute constant $c>0$ if $Q$ is quadratic. The latter bound may be reduced to $O(q^{-c'n}$) for some $c'>0$ when $Q(f)$ is a linear form in the coefficients of $f^2$, that is, a Hankel quadratic form, whereas for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.