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We give a necessary and sufficient condition for the integrality of theTaylor coefficients at the origin of formal power series $q_i({\mathbfz})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$, with ${\mathbf z}=(z_1,...,z_d)$and where $F({\mathbf z})$ and $G_i({\mathbf z})+\log(z_i)F({\mathbf z})$,$i=1,...,d$ are particular solutions of certain A-systems of differentialequations. This criterion is based on the analytical properties of Landau'sfunction (which is classically associated with the sequences of factorialratios) and it generalizes the criterion in the case of one variable presentedin "Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor desapplications miroir" [J. Reine Angew. Math.]. One of the techniques used toprove this criterion is a generalization of a version of a theorem of Dwork onthe formal congruences between formal series, proved by Krattenthaler andRivoal in "Multivariate $p$-adic formal congruences and integrality of Taylorcoefficients of mirror maps" [arXiv:0804.3049v3, math.NT]. This criterioninvolves the integrality of the Taylor coefficients of new univariate mirrormaps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2,math.AG] by Almkvist, van Enckevort, van Straten and Zudilin.

Criterion for the integrality of the Taylor coefficients of mirror maps in several variables