On the diagonals of rational functions: the minimal number of variables (unabridged version)

TL;DR

This work proposes and tests a unifying conjecture connecting the minimal multivariate representation of a diagonal of a rational function to the highest logarithmic exponent in the associated differential equation: $N_v = n+2$, provided the operator is homomorphic to its adjoint and thus has a symplectic or orthogonal Galois group. By surveying lattice Green functions, Ising-model susceptibilities, and Calabi–Yau operators, the paper demonstrates that the same $N_v$–$n$ relationship and parity rules ($N_v$ even $ ightarrow SO$, $N_v$ odd $ ightarrow Sp$) hold across irreducible and certain reducible denominators $Q$. It also analyzes how factorization of $Q$ into two polynomials yields direct-sum or dressed-product operators that preserve the Nv framework, and it illustrates multiple representations via pullbacks and roots, always returning to five-variable ($N_v=5$) representations for the Calabi–Yau cases. While the results are conjectural, they are consistently supported by extensive examples, suggesting a deep connection between diagonal representations, monodromy, and algebraic structures in mathematical physics. Overall, the work offers a coherent lens to view diagonals of rational functions through their differential-algebraic and geometric underpinnings, with potential implications for the rationality/integrality of $n$-fold integrals in physics and combinatorics.

Abstract

From some observations on the linear differential operators occurring in the Lattice Green function of the d-dimensional face centred and simple cubic lattices, and on the linear differential operators occurring in the n-particle contributions to the magnetic susceptibility of the square Ising model, we forward some conjectures on the diagonals of rational functions. These conjectures are also in agreement with exact results we obtain for many Calabi-Yau operators, and many other examples related, or not related to physics. Consider a globally bounded power series which is the diagonal of rational functions of a certain number of variables, annihilated by an irreducible minimal order linear differential operator homomorphic to its adjoint. Among the logarithmic formal series solutions, at the origin, of this operator, call n the highest power of the logarithm. We conjecture that this diagonal series can be represented as a diagonal of a rational function with a minimal number of variables N_v related to this highest power n by the relation N_v = n +2. Since the operator is homomorphic to its adjoint, its differential Galois group is symplectic or orthogonal. We also conjecture that the symplectic or orthogonal character of the differential Galois group is related to the parity of the highest power n, namely symplectic for n odd and orthogonal for n even. We also sketch the case where the denominator of the rational function is not irreducible and is the product of, for instance, two polynomials. The analysis of the linear differential operators annihilating the diagonal of rational function where the denominator is the product of two polynomials, sheds some light on the emergence of such mixture of direct sums and products of factors. The conjecture N_v = n +2 still holds for such reducible linear differential operators.