We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation.
JMM would like to thank J-P. Allouche, C. Banderier, G. Christol, S. Fischler, T. Rivoal, J. Rocques and J. Sauloy , and J-A. Weil for fruitful discussions on differentially algebraic functions. JMM would like to thank D. Bertrand for providing a proof by Ehud Hrushovski, on the closure by composition of differentially algebraic functions. He also thanks C. Koutschan and M. Mezzarobba for discussions on zeros of D-finite functions. This work has been performed without any support of the ANR, the ERC, the MAE or any PES of the CNRS. AJG and IJ would like to thank the Australian Research Council for its support through grant DP140101110. IJ was supported by an award under the Merit Allocation Scheme of the NCI National facility at the ANU, where the bulk of the large scale numerical computations were performed. JP would like to thank the Centre of Excellence for Mathematics and Statistics of Complex Systems for their hospitality in June 2015, where this research was partially conducted. JP was supported by the National Science Foundation and the Australian Academy of Science under NSF award #1514825.