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Newton polygon
From Wikipedia, the free encyclopedia
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields.
In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring
- K[[X]],
over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms
- aXr
of the power series expansion solutions to equations
- P(F(X)) = 0
where P is a polynomial with coefficients in K[X], the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in
- K[[Y]]
with Y = X1/d for a denominator d corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d.
After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.
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A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots.
Let K be a local field with discrete valuation vK and let
![f(x) = a_nx^n + \cdots + a_1x + a_0 \in K[x].](../../../math/f/c/a/fca58c87316ca6e7b48656a06e041331.png)
Then the Newton polygon of f is defined to be the convex hull of the set of points
In non-jargon: plot all of these points Pi on the xy-plane, then starting at P0, draw a ray straight up parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P1, break the ray here and keep rotating the remaining ray until it hits P2... continue until the process reaches the point Pn; the resulting polygon (and its interior) is the Newton polygon.
A practical application of the Newton polygon comes from the following result:
Let
be the slopes of the line segments of the Newton polygon of f(x) (as defined above) arranged in increasing order, and let
be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points Pi and Pj then the length is j − i). Then for each integer 
, f(x) has exactly λκ roots with valuation − μk.
In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual roots, in an algebraic closure. This has aspects both of ramification theory and singularity theory. The valid inferences possible are to the valuations first of the power sums, by means of Newton's identities.