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 $Q\in {\mathbb{F}}_q[x_0,\dots ,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\dots ,x_{n-1}$ 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 $O_\epsilon \left( q^{(-\frac{1}{4}+\epsilon)n} \right)$forany$\epsilon >0$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 $O(q^{-c^{\mathrm{\prime }}n}$) for some $c^{\mathrm{\prime }}>0$ 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.