I read here (page 237) that the valuation of the index of a polynomial is equal to the number of integer points below its Newton polygon.
I am confused how this makes sense--the cited paper (this) just detailed an algorithm to calculate the index using the formula. In the introduction it mentioned Ore proving something along the lines of that but the cited papers seem to be in German.
Montes wrote his PhD thesis on this interesting topic.
Perhaps also you will find useful the paper On a theorem of Ore by Montes and Nart. If that paper doesn't fully answer your questions please get back to me.
You can also look in Cohen's book, "A Course in Computational Algebraic Number Theory," Section 6.2.1 for a brief discussion of Newton polygons in the context of computing maximal orders, i.e. integral bases.
The Guardia-Montes-Nart algorithms have practical applications for computing integral bases for number fields of large degree and discriminant (that otherwise would be hard to do with classical algorithms).