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Section: New Results

Diagonals of rational functions and selected differential Galois groups

Diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we exhaustively analyze inĀ [1] three (constrained) sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. The conclusion is that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We also discuss conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.