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Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) =\sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is thesum of divisor function. This was proven true by Ford, Luca and Pomerance in2010. We ask the analogous question of whether infinitely often we have$\varphi(F) = \sigma(G)$ where $F$ and $G$ are polynomials over some finitefield $\mathbb{F}_q$. We find that when $q\not=2$ or $3$, then this can onlytrivially happen when $F=G=1$. Moreover, we give a complete characterisation ofthe solutions in the case $q=2$ or $3$. In particular, we show that $\varphi(F)= \sigma(G)$ infinitely often when $q=2$ or $3$.

On Incidences of $\varphi$ and $\sigma$ in the Function Field Setting