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.

  • $\begingroup$ If the crux of your Question is making sense of a specific paper, giving a citation or link to that paper would help your Readers. $\endgroup$ – hardmath Jul 19 '16 at 3:40
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    $\begingroup$ @hardmath I have added context to my question though the problem isn't really understanding the paper but finding a paper on the topic. $\endgroup$ – user354925 Jul 19 '16 at 4:03
  • $\begingroup$ 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? $\endgroup$ – hardmath Jul 19 '16 at 14:39

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).


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