Non-vanishing of primitive cubic, quartic and sextic L-functions

We establish lower bounds for the non-vanishing proportions of primitive cubic, quartic and sextic Dirichlet $L$-functions over ${\text{F}}_q[T]$ in the non-Kummer setting. Adapting the truncated-exponential method of Gao and Zhao, we improve the cubic lower bound of David, Florea and Lalín and obtain positive lower bounds for quartic and sextic families in which every ramified local character has full order. For each family, the lower limit of the non-vanishing proportion exceeds $e^{-e^{66}}$ as the conductor degree tends to infinity with $q$ fixed. We also establish uniform twisted first moments in the quartic and sextic cases, yielding an asymptotic formula for the sextic first moment. All results are unconditional.