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.^{[1]}^{[2]}

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.^{[3]} 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 2*n* variables.

For formal exponential polynomials over a field *K* we proceed as follows.^{[4]} Let *W* be a finitely generated **Z**-submodule of *K* and consider finite sums of the form

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*.^{[5]}^{[6]}

## Properties

**Ritt's theorem** states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.^{[4]}

## 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.

## Notes

**^**C. J. Moreno,*The zeros of exponential polynomials*, Compositio Mathematica 26 (1973), pp.69–78.**^**M. Waldschmidt,*Diophantine approximation on linear algebraic groups*, Springer, 2000.**^**Martin Bays, Jonathan Kirby, A.J. Wilkie,*A Schanuel property for exponentially transcendental powers*, (2008), arXiv:0810.4457v1- ^
^{a}^{b}Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003).*Recurrence sequences*. Mathematical Surveys and Monographs.**104**. Providence, RI: American Mathematical Society. p. 140. ISBN 0-8218-3387-1. Zbl 1033.11006. **^**László Székelyhidi,*On the extension of exponential polynomials*, Mathematica Bohemica**125**(2000), pp.365–370.**^**P. G. Laird,*On characterizations of exponential polynomials*, Pacific Journal of Mathematics**80**(1979), pp.503–507.