Newton polygon

Construction of the Newton polygon of the polynomial

P(X) = 1 + 5X + \frac{1}{5}X^2 + 35X^3 + 25X^5 + 625X^6

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

aX^r

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 = X^1/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.

Definition

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 $v_K$ and let

f(x) = a_nx^n + \cdots + a_1x + a_0 \in K[x]

with $a_0 a_n \ne 0$ . Then the Newton polygon of $f$ is defined to be the lower convex hull of the set of points

P_i=\left(i,v_K(a_i)\right),

ignoring the points with $a_i = 0$ . Restated geometrically, plot all of these points P_i on the xy-plane. Let's assume that the points indices increase from left to right (P₀ is the leftmost point, P_n 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 P_k₁ (not necessarily P₁). Break the ray here. Now draw a second ray from P_k₁ straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P_k₂. Continue until the process reaches the point P_n; the resulting polygon (containing the points P₀, P_k₁, P_k₂, ..., P_{k_m}, P_n) is the Newton polygon.

Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P₀, ..., P_n. 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 Ch6 §3 of "Local Fields" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition.

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.^[1]

Applications

A Newton Polygon is sometimes a special case of a Newton Polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like $3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y = 0$

This diagram shows the Newton polygon for P(x,y) = 3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y, with positive monomials in red and negative monomials in cyan. Faces are labelled with the limiting terms they correspond to.

Another application of the Newton polygon comes from the following result:

Let

\mu_1, \mu_2, \ldots, \mu_r

be the slopes of the line segments of the Newton polygon of $f(x)$ (as defined above) arranged in increasing order, and let

\lambda_1, \lambda_2, \ldots, \lambda_r

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 $P_i$ and $P_j$ then the length is $j-i$ ). Then for each integer $1\leq\kappa\leq r$ , $f(x)$ has exactly $\lambda_{\kappa}$ roots with valuation $-\mu_{\kappa}$ .

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.

References

↑ Egbert Brieskorn, Horst Knörrer (1986). Plane Algebraic Curves, pp. 370–383.

Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131
Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199.

External links

Wikimedia Commons has media related to Newton polygon.

Applet drawing a Newton Polygon

Isaac Newton

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	Hypotheses non fingo Standing on the shoulders of giants

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose Newton (unit)

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