Direct cosmographic reconstruction of the quintessence potential

We derive expressions for the first and second derivatives of the quintessence potential $V(\phi )$, in terms of $\lambda =-V^{\mathrm{\prime }}\mathrm{/}V$ and $\mathrm{\Gamma }=(V^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}V)\mathrm{/}(V^{\mathrm{\prime }}\mathrm{/}V{)}^2$, as functions of the quintessence density fraction ${\mathrm{\Omega }}_{\phi }$ and the cosmographic parameters $q$, $j$, and $s$. Our mapping is not explicitly a function of the equation of state parameter $w$. We use these results, along with recent observational data, to derive expansions of $V(\phi )$ about the present-day value of the scalar field, ${\phi }_0$.