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. 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 ex. 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. 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 where the fi are polynomials in K[X] and the exp(wi X) are formal symbols indexed by wi in W subject to exp(u + v) = exp(u) exp(v). 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. Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.