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The enumeration of planar maps equipped with an Eulerian orientation hasattracted attention in both combinatorics and theoretical physics since atleast 2000. The case of 4-valent maps is particularly interesting: theseorientations are in bijection with properly 3-coloured quadrangulations, whilein physics they correspond to configurations of the ice model.  We solve both problems -- namely the enumeration of planar Eulerianorientations and of 4-valent planar Eulerian orientations -- by expressing theassociated generating functions as the inverses (for the composition of series)of simple hypergeometric series. Using these expressions, we derive theasymptotic behaviour of the number of planar Eulerian orientations, thusproving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann.This behaviour, $\mu^n /(n \log n)^2$, prevents the associated generatingfunctions from being D-finite. Still, these generating functions aredifferentially algebraic, as they satisfy non-linear differential equations oforder $2$. Differential algebraicity has recently been proved for other mapproblems, in particular for maps equipped with a Potts model.  Our solutions mix recursive and bijective ingredients. In particular, apreliminary bijection transforms our oriented maps into maps carrying a heightfunction on their vertices. In the 4-valent case, we also observe an unexpectedconnection with the enumeration of maps equipped with a spanning tree that isinternally inactive in the sense of Tutte. This connection remains to beexplained combinatorially.

The generating function of planar Eulerian orientations