  In recent decades, the use of ideas from Minkowski's Geometry of Numbers has
gained recognition as a helpful tool in bounding the number of solutions to
modular congruences with variables from short intervals. In 1941, Mahler
introduced an analogue to the Geometry of Numbers in function fields over
finite fields. Here, we build on Mahler's ideas and develop results useful for
bounding the sizes of intersections of lattices and convex bodies in
$\mathbb{F}_q((1/T))^d$, which are more precise than what is known over
$\mathbb{R}^d$. These results are then applied to various problems regarding
bounding the number of solutions to congruences in $\mathbb{F}_q[T]$, such as
the number of points on polynomial curves in low dimensional subspaces of
finite fields. Our results improve on a number of previous bounds due to
Bagshaw, Cilleruelo, Shparlinski and Zumalac\'{a}rregui. We also present
previous techniques developed by various authors for estimating certain
energy/point counts in a unified manner.
