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.