Newton Polygon
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Newton polygon is a tool for understanding the behaviour of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s over
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s, or more generally, over ultrametric fields. In the original case, the local field of interest was ''essentially'' the field of
formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums ...
in the indeterminate ''X'', i.e. the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
ring K X, over K, where K was the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
field. This is still of considerable utility with respect to
Puiseux expansion In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
s. 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 /math>, the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
; that is, implicitly defined
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additio ...
s. The exponents r here are certain
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, depending on the
branch A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
chosen; and the solutions themselves are power series in K Y with Y = X^ 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 number In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
s, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. Newton polygons have also been useful in the study of
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s.


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
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
endowed with a non-archimedean valuation v_K: K \to \mathbb R\cup \, and let :f(x) = a_nx^n + \cdots + a_1x + a_0 \in K with a_0 a_n \ne 0. Then the Newton polygon of f is defined to be the lower boundary of the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
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 (''P''''0'' is the leftmost point, ''P''''n'' is the rightmost point). Then, starting at ''P''0, 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 ''P''1). 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 ''P''0, ''P''k1, ''P''k2, ..., ''P''km, ''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 ''P''0, ..., ''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.


Main theorem

With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, which states that the valuation of the roots of f are entirely determined by its Newton polygon: 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 segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s 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). * The \mu_i are distinct; * \sum_i \lambda_i = n; * if \alpha is a root of f in K, v(\alpha) \in \; * for every i, the number of roots of f whose valuations are equal to -\mu_i (counting multiplicities) is at most \lambda_i, with equality if f splits into the product of linear factors over K.


Corollaries and applications

With the notation of the previous sections, we denote, in what follows, by L the splitting field of f over K, and by v_L an extension of v_K to L. Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example: * ''Suppose that the valuation v is discrete and normalized, and that the Newton polynomial of f contains only one segment whose slope is \mu and projection on the x-axis is \lambda. If \mu = a/n, with a coprime to n, then f is irreducible over K. In particular, since the Newton polygon of an Eisenstein polynomial consists of a single segment of slope -\frac connecting (0,1) and (n,0),
Eisenstein criterion In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials wi ...
follows.'' Indeed, by the main theorem, if \alpha is a root of f, v_L(\alpha) = -a/n. If f were not irreducible over K, then the degree d of \alpha would be < n , and there would hold v_L(\alpha) \in \mathbb Z. But this is impossible since v_L(\alpha) = -a/n with a coprime . Another simple corollary is the following: * ''Assume that (K, v_K) is
Henselian In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...
. If the Newton polygon of f fulfills \lambda_i = 1 for some i, f has a root in K.'' ''Proof:'' By the main theorem, f must have a single root \alpha whose valuation is v_L(\alpha) = -\mu_i. In particular, \alpha is separable over K. If \alpha does not belong to K, \alpha has a distinct Galois conjugate \alpha' over K, with v_L(\alpha') = v_L(\alpha), and \alpha' is a root of f, a contradiction. More generally, the following factorization theorem holds: * ''Assume that (K, v_K) is
Henselian In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...
. Then f = A\,f_1\, f_2\cdots f_r,, where A\in K, f_i\in K /math> is monic for every i, the roots of f_i are of valuation -\mu_i , and \deg(f_i) = \lambda_i.'' :''Moreover, \mu_i = v_K(f_i(0))/\lambda_i, and if v_K(f_i(0)) is coprime to \lambda_i, f_i is irreducible over K.'' ''Proof:'' For every i, denote by f_i the product of the monomials (X - \alpha) such that \alpha is a root of f and v_L(\alpha) = -\mu_i. We also denote f = A P_1^P_2^\cdots P_s^ the factorization of f in K /math> into prime monic factors (A\in K). Let \alpha be a root of f_i. We can assume that P_1 is the minimal polynomial of \alpha over K. If \alpha' is a root of P_1, there exists a K-automorphism \sigma of L that sends \alpha to \alpha', and we have v_L(\sigma \alpha) = v_L(\alpha) since K is Henselian. Therefore \alpha' is also a root of f_i. Moreover, every root of P_1 of multiplicity \nu is clearly a root of f_i of multiplicity k_1\nu, since repeated roots share obviously the same valuation. This shows that P_1^ divides f_i. Let g_i = f_i/P_1^. Choose a root \beta of g_i. Notice that the roots of g_i are distinct from the roots of P_1. Repeat the previous argument with the minimal polynomial of \beta over K, assumed w.l.g. to be P_2, to show that P_2^ divides g_i. Continuing this process until all the roots of f_i are exhausted, one eventually arrives to f_i = P_1^\cdots P_m^, with m \leq s. This shows that f_i\in K /math>, f_i monic. But the f_i are coprime since their roots have distinct valuations. Hence clearly f = A f_1\cdot f_2\cdots f_r, showing the main contention. The fact that \lambda_i = \deg(f_i) follows from the main theorem, and so does the fact that \mu_i = v_K(f_i(0))/\lambda_i, by remarking that the Newton polygon of f_i can have only one segment joining (0, v_K(f_i(0)) to (\lambda_i, 0 = v_K(1)). The condition for the irreducibility of f_i follows from the corollary above. (q.e.d.) The following is an immediate corollary of the factorization above, and constitutes a test for the reducibility of polynomials over Henselian fields: * ''Assume that (K, v_K) is
Henselian In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...
. If the Newton polygon does not reduce to a single segment (\mu, \lambda), then f is reducible over K.'' Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a
Newton polytope In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, give ...
, 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.


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 In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...
. This has aspects both of
ramification theory In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
and
singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
. The valid inferences possible are to the valuations of power sums, by means of
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
.


History

Newton polygons are named after
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
, who first described them and some of their uses in correspondence from the year 1676 addressed to
Henry Oldenburg Henry Oldenburg (also Henry Oldenbourg) FRS (c. 1618 as Heinrich Oldenburg – 5 September 1677), was a German theologian, diplomat, and natural philosopher, known as one of the creators of modern scientific peer review. He was one of the fo ...
.
Egbert Brieskorn Egbert Valentin Brieskorn (7 July 1936, in Rostock – 11 July 2013, in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution. Education Brieskorn was born in 1936 as the son of a mill cons ...
,
Horst Knörrer Horst Knörrer (born 31 July 1953, in Bayreuth) is a German mathematician, who studies algebraic geometry and mathematical physics. Knörrer studied from 1971 at University of Regensburg and University of Erlangen-Nuremberg and received a doctor ...
(1986). ''Plane Algebraic Curves'', pp. 370–383.


See also

*
F-crystal In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter ''F'' stands for Frobenius, indicating that ''F''-crystals have an action of Frobenius on them. F-isocrys ...
*
Eisenstein's criterion In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with ...
*
Newton–Okounkov body In algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a convex body in Euclidean space associated to a divisor (or more generally a linear system) on a variety. The convex geometry of a Newton–Okounkov body encodes (as ...
*
Newton polytope In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, give ...


References

* * Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199.


External links


Applet drawing a Newton Polygon
{{Isaac Newton Algebraic number theory Symmetric functions Isaac Newton