Valuation of Index of polynomial with Newton Polygon

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.

• Looking at page 237 from the book, "Algorithmic Arithmetic, Geometry, and Coding Theory," there is a mention at the bottom of the page of "how to compute $ind(f)$ as the accumulation of the number of points of integer coordinates lying below all Newton polygons that occur along the flow of the Montes algorithm." Is this the topic about which you are trying to find a paper or other exposition? – hardmath Jul 19 '16 at 14:39