Title: On the diagonals of rational functions: the minimal number of variables (unabridged version) Show Chinese title

Title: 关于有理函数的对角线：变量的最小数量（未删节版）

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. Show Chinese abstract

Abstract: 基于对出现在d维面心立方和简单立方格点的晶格Green函数中的线性微分算子以及出现在方形Ising模型磁化率的n粒子贡献中的线性微分算子的一些观察，我们提出了关于有理函数对角线的一些猜想。这些猜想也与我们在许多Calabi-Yau算子及相关或不相关的许多其他例子中获得的精确结果一致。 考虑一个全局有界的幂级数，它是某些变量的有理函数的对角线，并被一个不可约的最小阶线性微分算子所消去，且该算子与它的伴随算子同态。在该算子在原点处的对数形式解中，记log的最高次幂为n。 我们猜测这个对角线级数可以表示为具有最少变量数N_v的有理函数的对角线，其中N_v与最高次幂n的关系由N_v = n + 2给出。 由于算子与它的伴随算子同态，其微分Galois群是辛群或正交群。 我们还猜测微分Galois群的辛或正交特性与最高次幂n的奇偶性有关，即当n为奇数时为辛，当n为偶数时为正交。 我们还概述了分母不是不可约的情况，例如分母是两个多项式的乘积的情况。 分析使有理函数对角线被消去的线性微分算子，有助于理解直接和与因子乘积混合出现的原因。 对于这种可约的线性微分算子，猜想N_v = n + 2仍然成立。