Differential analogue of the newton polygon

Question

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows:

Let us recall the definition of the Newton polygon at $\infty$ of an equation $$p(x)z''+q(x)z'+r(x)z=0$$ where $p,q,r$ are polynomials. This will be the convex hull with positive slopes of the three points $$(0,-d^\circ r),(1,-d^\circ q+1), (2, -d^\circ p+2)$$

I'm wondering whether anyone has seen this before and could possibly explain the general method behind this? To me it seems as if it gives the conditions for a differential equation to have a closed form analytic solution. Also, the notation is a bit confusing; I'm not aware of what the $d^\circ$ means.

Answer 1

[This should rather be a comment, but I have not yet enough reputation for comments, so I post it as an answer.]

I do not have access now to the book you are citing and therefore cannot properly compare the definitions, but Newton polygons are used in the classification of linear differential equations. You can an introduction in 3.3 of M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, Volume 328, Springer, 2003.