French-Australian International Emerging Action on Physics


Dr. Jean-Marie Maillard


Zeroes of the partition function of the square Ising model in a magnetic field.

Equimodular curves of the square Ising model in a magnetic field.

Zeroes of the partition function of the square Ising model in a magnetic field.

Singularities of D-finite n-fold integrals associated to the suceptibility of the square Ising model.

Effective algebraic geometry: rational points of an algebraic surface in 15 dimensions generated by birational symmetries.

Effective algebraic geometry: rational points of an algebraic surface in 15 dimensions generated by birational symmetries.


The IEA SERINT managed by Dr Jean-Marie Maillard (LPTMC, UMR 7600, CNRS – Sorbonne Université, Paris) in collaboration with the Department of Mathematics & Statistics, The University of Melbourne, Parkville Vic. Australia (Contact: Prof. A. J. Guttmann) is effective since 2018.

    Missions and research themes

    Series with integer coefficients emerge quite naturally in lattice statistical mechanics and enumerative combinatorics, the celebrated two-dimensional Ising model being the perfect illustration of this occurrence.  One remarks that such series are quite often solutions of linear and non-linear differential equations and have quite intriguing mathematical properties corresponding to different domain of mathematics (arithmetic, number theory, differential algebra, algebraic geometry, …).  Along this line, we had shown that the n-fold integrals occurring in theoretical physics are naturally diagonals of rational functions and this explains most of the intriguing and quite unexpected properties of so many physical quantities (reduction of the corresponding series with integer coefficients to algebraic functions modulo primes or power of primes, the series being solutions of globally nilpotent operators, …).

    The goal of IEA SERINT is to understand the intriguing mathematical properties of such series with integer coefficients emerging in theoretical physics. In particular we want to see if such series are essentially associated with some integrability properties of the models, or correspond to a larger framework. More precisely, we want to see if they mostly correspond to (holonomic, D-finite) linear differential equations, if they can correspond to selected (differentially algebraic) non-linear differential equations, or if they can  also  correspond to a much larger (differentially transcendental) framework. Consequently, the IEA SERINT is naturally organised according to these miscellaneous “dualities”: linear versus non-linear differential equations, integrability versus non-integrability, differentially algebraic versus differentially transcendental.

    It is also organised according to the various  well-suited mathematical domains and tools: differential algebra (creative telescoping approach), formal calculations  (DEtools in Maple, gfun), effective algebraic geometry and mod. prime calculations, analytic calculations (random theory, Stieltjes series), etc …


    We first obtained results in the simplest holonomic (i.e. linear) framework, revisiting a large set of non trivial results for diagonals of rational functions and series solutions of “telescopers” with a more intrinsic effective algebraic geometry approach. This sheds some light on the frequent and remarkable occurrence of modular forms in so many non-trivial examples in theoretical physics.

    We also made some important progress on the so-called “Christol’s conjecture” which amounts to conjecturing that any holonomic series with integer coefficients, and a finite radious of convergence, is a diagonal of a rational function. In a more enumerative combinatorics framework we revisited the occurrence of Stieltjes series, namely series with positive integer coefficients which can be seen as moments of an underlying measure, considering selected examples of enumerative combinatorics (pattern avoiding permutations). The exact calculations we performed in a holonomic (sometimes modular form) framework shed some light on relations of such series with positive integer coefficients  with various mathematical structures (random matrix theory, orthogonal polynomials, Hankel/Toeplitz matrices, …). 

    One notes that these relations may also work in a much larger non-holonomic framework. We still need to get results corresponding to a deeper understanding of differentially algebraic series with integer coefficients. Along this line we have already obtained some quite puzzling results for two-point correlation functions of the square Ising model: for a particular subcase of these two-point correlation functions we have seen  they are actually solutions of a two-parameter family of non-linear ODEs with the Painlevé property (fixed critical points), which corresponds exactly to sigma forms of Painlevé VI.

    A lot of work remains to be done in the general cases. Can we simply describe the non-linear ODEs with the Painlevé property for the general  two-point correlation funtions ? Are the corresponding series Stieltjes series ? Do the corresponding series reduce to algebraic functions modulo primes, or power of primes ?

    institutions and laboratories involved