Any Facts about Newton Polygons

In summary, the speaker is looking for short and interesting uses of Newton Polygons for polynomials to discuss in a upcoming talk. They mention having a cool result about irreducibility and using them to look at polynomials in the p-adics, but also mention that they have some uses in differential equations. They provide a link to a resource that may be helpful.
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snoble
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I'm giving a talk in a few days about Newton Polygons for polynomials and I was wondering if anyone knew of a few (short) interesting uses I could discuss or perhaps a text with a chapter on the subject. I have a cool result about irreducibility and of course the basics but it would be nice to have a few smaller facts that I can fill 10 or 20 minutes with.

I'm using them to look at polynomials in the p-adics but I understand that they have some uses in DE's as well although I don't know much about them.

Thanks,
Steven
 
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Related to Any Facts about Newton Polygons

What are Newton polygons?

Newton polygons are a type of graphical representation used in mathematics to study the roots of polynomials, or algebraic equations with multiple terms. They were first introduced by Sir Isaac Newton in the 17th century.

How are Newton polygons constructed?

To construct a Newton polygon, the polynomial in question is first written in descending order of its exponents. Then, the differences between the exponents of consecutive terms are plotted on a graph, with the x-axis representing the exponents and the y-axis representing the coefficients.

What is the significance of the slopes in a Newton polygon?

The slopes of the segments in a Newton polygon represent the roots of the polynomial. A segment with a slope of 1 corresponds to a root of multiplicity 1, a slope of 2 corresponds to a root of multiplicity 2, and so on.

What can be determined from a Newton polygon?

From a Newton polygon, one can determine the number of roots of a polynomial, their multiplicities, and their approximate values. It can also be used to determine the degree of a polynomial and to factor it into linear factors.

Are Newton polygons still used in modern mathematics?

Yes, Newton polygons are still used in modern mathematics, particularly in the study of algebraic equations and their roots. They are also used in other fields such as physics and engineering to analyze and solve problems involving equations with multiple terms.

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