Exponential polynomial
# Exponential polynomial

In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.

## Definition

### In fields

An exponential polynomial generally has 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 $e^x.$ In this setting the term exponential polynomial is often used to mean polynomials of the form P(x, $e^x)$ 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 $e^x$ above. Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form $P(x_1,$ ..., $x_n,$ $e^{x1},$ ..., $e^{xn}),$ where P is a polynomial in 2n variables.

For formal exponential polynomials over a field K we proceed as follows. Let W be a finitely generated Z-submodule of K and consider finite sums of the form

    $\sum _{i=1}^{m}f_{i}(X)\exp(w_{i}X)\ ,$

where the $f_i$ are polynomials in K[X] and the $exp(w_i$ X) are formal symbols indexed by $w_i$ in W subject to exp(u + v) = exp(u) exp(v).

### 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 group 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.

## Properties

Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.

## Applications

Exponential polynomials on R and C often appear in transcendental number theory, where they appear as auxiliary functions 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 $R^n$ 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.

Exponential polynomials appear in the characteristic equation associated with linear delay differential equations.

