On the D-finiteness of generating functions counting small steps walks in the quadrant

The enumeration of small steps walks confined to the first quadrant of the plane has attracted a lot of attention over the past fifteen years. The associated generating functions are trivariate formal power series in $x,y,t$ where the parameter $t$ encodes the length of the walk while the variables $x,y$ correspond to the coordinates of its ending point. These functions satisfy a functional equation in two catalytic variables. Bousquet-Mélou and Mishna have associated to any small steps model an algebraic curve called the kernel curve and a group called the group of the walk. These two objects turned out to be central in the classification of small steps models. In a recent work, Dreyfus, Elvey Price, and Raschel prove that the group of the walk is finite if and only if the generating function is $D$-finite, that is, it satisfies a linear differential equation with polynomial coefficients in each of its variables $x,y,t$. In this paper, we show that if the group of the walk is infinite, the generating function doesn't satisfy a linear differential equation in $x,y$ or $t$ over the field $\mathbb{Q}(x,y,t)$. The proof of Dreyfus, Elvey Price, and Raschel is based on some singularity analysis. Here, we propose a new strategy which relies essentially on the aforementioned functional equation and on algebraic arguments. This point of view sheds also a new light on the algebraic nature of the generating functions of small steps models since it relates their $D$-finiteness more directly to some geometric properties of the kernel curve.