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Wikipedia / Newton polygon

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 ultrametric 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 was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions of the power series, the polynomial ring; that is, implicitly defined algebraic functions here are certain rational numbers, depending on the branch.

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.

Definition

A priori, given a polynomial over a field, the behaviour of the rootss (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots.

be a field endowed with a non-archimedean valuation, and let

is defined to be the lower boundary of the convex hull.

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. Now draw a second ray from Pk1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk2. Continue until the process reaches the point Pn; the resulting polygon (containing the points P0, Pk1, Pk2, ..., Pkm, Pn) is the Newton polygon.

Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points.

For a neat diagram of this see

Main theorem

With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, are entirely determined by its Newton polygon:

be the corresponding lengths of the line segments).

  • are distinct;

  • ;

  • ;

  • .

Corollaries and applications

.

The Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example:

coprime .

Another corollary is the following:

More generally, the following factorization theorem holds:

The following constitutes a test for the reducibility of polynomials over Henselian fields:

Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a Newton polytope

Symmetric function explanation

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 of power sums, by means of Newton's identities.

History

Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg.

See also

Notes

References