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Concept
Newton polygon
Summary
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields.
In the original case, the local field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring ,
over , where 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
of the power series expansion solutions to equations
where is a polynomial with coefficients in , the polynomial ring; that is, implicitly defined algebraic functions. The exponents here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in
with for a denominator corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating .
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.
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 be a field endowed with a non-archimedean valuation , and let
with . Then the Newton polygon of is defined to be the lower boundary of the convex hull of the set of points
ignoring the points with .
Restated geometrically, plot all of these points Pi on the xy-plane. Let's assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P0, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk1 (not necessarily P1). Break the ray here.
Official source
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