Abstract
Thakur [Tha10] showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $\zeta_A(r)$ and $\zeta_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $\zeta_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct an $\mathbb{F}_q[t]$-module $X$. To determine effectively whether an $n$-tuple of coefficients in $\mathbb{F}_q(\theta)$ satisfies a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $v\in X$ constructed with respect to the given coefficients. We also provide an effective criterion for the $\mathbb{F}_q[t]$-torsion property of the point $v$.