Diophantine approximation of polynomials over ${\mathbb{F}}_q[t]$ satisfying a divisibility condition

Let ${\mathbb{F}}_q[t]$ denote the ring of polynomials over ${\mathbb{F}}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over ${\mathbb{F}}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.