A note on Diophantine inequalities in function fields

We will discuss how the Bentkus–Götze–Freeman variant of the Davenport–Heilbronn circle method can be used to study $\mathbb{F}_q[t]$ solutions to inequalities of the form \[\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k-\gamma)<\tau , \] where constants $\lambda_1,\dots, \lambda_s \in\mathbb{F}_q((1/t))$ satisfy certain conditions. This result is a generalization of the work done by Spencer in [11] to count the number of solutions to inequalities of the form \[\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k)<\tau.\]