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DP-coloring (or correspondence coloring) is a generalization of list coloringthat has been widely studied since its introduction by Dvo\v{r}\'{a}k andPostle in 2015. As the analogue of the chromatic polynomial of a graph $G$,$P(G,m)$, and the list color function, $P_{\ell}(G,m)$, the DP color functionof $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-coloringsover all possible $m$-fold covers. It follows that $P_{DP}(G,m) \leP_{\ell}(G,m) \le P(G,m)$. A function $f$ is chromatic-adherent if for everygraph $G$, $f(G,a) = P(G,a)$ for some $a \geq \chi(G)$ implies that $f(G,m) =P(G,m)$ for all $m \geq a$. It is known that the DP color function is notchromatic-adherent, but there are only two known graphs that demonstrate this.Suppose $G$ is an $n$-vertex graph and $\mathcal{H}$ is a 3-fold cover of $G$,in this paper we associate with $\mathcal{H}$ a polynomial $f_{G, \mathcal{H}}\in \mathbb{F}_3[x_1, \ldots, x_n]$ so that the number of non-zeros of $f_{G,\mathcal{H}}$ equals the number of $\mathcal{H}$-colorings of $G$. We then usea well-known result of Alon and F\"{u}redi on the number of non-zeros of apolynomial to establish a non-trivial lower bound on $P_{DP}(G,3)$ when $2n >|E(G)|$. An easy consequence of this is that $P_{DP}(G, 3) \geq 3^{n/6}$ forevery $n$-vertex planar graph $G$ of girth at least 5, improving the previouslyknown bounds on both $P_{DP}(G, 3)$ and $P_{\ell}(G, 3)$. Finally, we use thisbound to show that there are infinitely many graphs that demonstrate thenon-chromatic-adherence of the DP color function.

An Algebraic Approach for Counting DP-3-colorings of Sparse Graphs