Newton polygon
Encyclopedia
In mathematics
, the Newton polygon is a tool for understanding the behaviour of polynomial
s over local field
s.
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
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
of the power series expansion solutions to equations
where P is a polynomial with coefficients in K[X], the polynomial ring
; that is, implicitly defined
algebraic function
s. The exponents r here are certain rational number
s, depending on the branch chosen; and the solutions themselves are power series in
with Y = X1/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 number
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
. Newton polygons have also been useful in the study of elliptic curve
s.
Let
be a local field
with discrete valuation
and let
with
. Then the Newton polygon of 
is defined to be the lower 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. 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.
Let
be the slopes of the line segments of the Newton polygon of
(as defined above) arranged in increasing order, and let
be the corresponding lengths of the line segment
s projected onto the x-axis (i.e. if we have a line segment stretching between the points
and 
then the length is 
). Then for each integer
, 
has exactly 
roots with valuation 
.
. This has aspects both of ramification theory and singularity theory
. The valid inferences possible are to the valuations first of the power sum
s, by means of Newton's identities
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Newton polygon is a tool for understanding the behaviour of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s over local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s.
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
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
of the formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
ring
- K
X ,XX is the twenty-fourth letter in the basic modern Latin alphabet.-Uses:In mathematics, x is commonly used as the name for an independent variable or unknown value. The usage of x to represent an independent or unknown variable can be traced back to the Arabic word šay شيء = “thing,” used in Arabic...
over K, where K was the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms
- aXr
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
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
; that is, implicitly defined
Implicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s. The exponents r here are certain rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, depending on the branch chosen; and the solutions themselves are power series in
- K
Y YY is the twenty-fifth letter in the basic modern Latin alphabet and represents either a vowel or a consonant in English.-Name:In Latin, Y was named Y Graeca "Greek Y". This was pronounced as I Graeca "Greek I", since Latin speakers had trouble pronouncing , which was not a native sound...
with Y = X1/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 number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s, it was shown that the Newton polygon is just as useful in questions of ramification
Ramification
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 local fields, and hence in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
. Newton polygons have also been useful in the study of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
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
be a local fieldLocal field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
with discrete valuation
and letwith
. Then the Newton polygon of is defined to be the lower convex hullConvex hull
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....
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. 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.
Applications
A practical application of the Newton polygon comes from the following result:Let
be the slopes of the line segments of the Newton polygon of
(as defined above) arranged in increasing order, and letbe the corresponding lengths of the line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
s projected onto the x-axis (i.e. if we have a line segment stretching between the points
and then the length is ). Then for each integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
, has exactly roots with valuation .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 closureAlgebraic 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....
. This has aspects both of ramification theory and singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
. The valid inferences possible are to the valuations first of the power sum
Power sum
The topic of power sums is treated at:* Power sum symmetric polynomial * Newton's identities * Symmetric function * Faulhaber's formula...
s, by means of Newton's identities
Newton's identities
In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...
.