Linear relations among algebraic points on tensor powers of the Carlitz module

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual $t$-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence for Carlitz polylogarithms at algebraic points in both $\mathrm{\infty }$-adic and $v$-adic settings.