Thakur (2010) 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 a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{ C}$ in $\mathbb{ F}_q(\theta)^{ r+s}$ gives a shuffle relation, we relate it to the $\mathbb{ F}_q[t]$-torsion property of the point $\mathbf{ v}_\mathfrak{ C}\in E'(\mathbb{ F}_q[\theta])$ constructed with respect to the given $(r+s)$-tuple $\mathfrak{ C}$. We also provide an effective criterion for deciding the $\mathbb{ F}_q[t]$-torsion property of the point $\mathbf{ v}_\mathfrak{ C}$.
Categories: math.NT