We examine correlations of the Möbius function over
$\mathbb{F}_{q}[t]$
with linear or quadratic phases, that is, averages of the form 1
$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f<n}\unicode[STIX]{x1D707}(f)\unicode[STIX]{x1D712}(Q(f))\end{eqnarray}$$
for an additive character
$\unicode[STIX]{x1D712}$
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
$f=\sum _{i<n}x_{i}t^{i}$
. As in the integers, it is reasonable to expect that, due to the random-like behaviour of
$\unicode[STIX]{x1D707}$
, such sums should exhibit considerable cancellation. In this paper we show that the correlation (
1) is bounded by
$O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$
for any
$\unicode[STIX]{x1D716}>0$
if
$Q$
is linear and
$O(q^{-n^{c}})$
for some absolute constant
$c>0$
if
$Q$
is quadratic. The latter bound may be reduced to
$O(q^{-c^{\prime }n})$
for some
$c^{\prime }>0$
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.