Exponential polynomial
Encyclopedia
In mathematics
, exponential polynomials are functions on fields
, rings
, or abelian group
s that take the form of polynomial
s in a variable and an exponential function
.
x. In this setting the term exponential polynomial is often used to mean polynomials of the form P(x,ex) where P ∈ C[x,y] is a polynomial in two variables.
There is nothing particularly special about C here, exponential polynomials may also refer to such a polynomial on any exponential field
or exponential ring with its exponential function taking the place of ex above. Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form P(x1,...,xn,ex1,...,exn), where P is a polynomial in 2n variables.
G a homomorphism
from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.
, where they appear as auxiliary function
s in proofs involving the exponential function. They also act as a link between model theory
and analytic geometry
. If one defines an exponential variety to be the set of points in Rn where some finite collection of exponential polynomials vanish, then results like Khovanskiǐ's theorem in differential geometry and Wilkie's theorem
in model theory show that these varieties are well-behaved in the sense that the collection of such varieties is stable under the various set-theoretic operations as long as one allows the inclusion of the image under projections of higher-dimensional exponential varieties. Indeed, the two aforementioned theorems imply that the set of all exponential varieties forms an o-minimal structure over R.
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...
, exponential polynomials are functions on fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, or abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s that take the form 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 in a variable and an exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
.
In fields
There is no single definition of what an exponential polynomial is, but the term is generally used to describe polynomials in both a variable x and some kind of exponential function E(x). In the complex numbers there is already a canonical exponential function, the function that maps x to eE (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
x. In this setting the term exponential polynomial is often used to mean polynomials of the form P(x,ex) where P ∈ C[x,y] is a polynomial in two variables.
There is nothing particularly special about C here, exponential polynomials may also refer to such a polynomial on any exponential field
Exponential field
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.-Definition:...
or exponential ring with its exponential function taking the place of ex above. Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form P(x1,...,xn,ex1,...,exn), where P is a polynomial in 2n variables.
In abelian groups
A more general framework where the term exponential polynomial may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian groupTopological abelian group
In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative....
G a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.
Uses
Exponential polynomials on R and C often appear in transcendence theoryTranscendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, where they appear as auxiliary function
Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high...
s in proofs involving the exponential function. They also act as a link between model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
. If one defines an exponential variety to be the set of points in Rn where some finite collection of exponential polynomials vanish, then results like Khovanskiǐ's theorem in differential geometry and Wilkie's theorem
Wilkie's theorem
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.-Formulations:...
in model theory show that these varieties are well-behaved in the sense that the collection of such varieties is stable under the various set-theoretic operations as long as one allows the inclusion of the image under projections of higher-dimensional exponential varieties. Indeed, the two aforementioned theorems imply that the set of all exponential varieties forms an o-minimal structure over R.