NationMaster - Encyclopedia: Newton polygon

In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...

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 In mathematics, a Laurent series is an infinite series. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

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 In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a Puiseux expansion is a formal power series expansion of an algebraic function. ...

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 In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ... In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non-trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

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. The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

1 Definition
2 Applications
3 Symmetric function explanation
4 See also

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 In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In mathematics, a discrete valuation on a commutative ring A is a function satisfying the conditions . For example, if A is the ring of integers, these properties are satisfied with ν(n) the largest value of k such that 2k divides n. ...

$f(x) = a_nx^n + ldots + a_1x + a_0 in K[x].$

Then the Newton polygon of $f$ is defined to be the convex hull of the set of points Convex Hull: Elastic band analogy // Alternative definitions In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. (Note that X may be the union of any set of objects made of points). ...

$P_i=left(i,v_K(a_i)right).$

In non-jargon: plot all of these points P_i 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 P_n; the resulting polygon (and its interior) is the Newton polygon. In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric...

Applications

A practical 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 $1leqkappaleq r$ , $f (x)$ has exactly $λ κ$ roots with valuation $μ k$ . In mathematics, a line segment is a part of a line that is bounded by two end points. ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

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 first of the power sums, by means of Newton's identities. In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... For non-mathematical singularity theories, see singularity. ... In mathematics, Newtons identities relate two different ways of describing the roots of a polynomial. ... In mathematics, Newtons identities relate two different ways of describing the roots of a polynomial. ...

Contents

Applications

Symmetric function explanation

See also