Exponential polynomial: Difference between revisions
Defined over fields and abelian groups, mentioned uses in transcendence and model theory |
wlink |
||
Line 1: | Line 1: | ||
{{Otheruses4|polynomials in variables and exponential functions|the polynomials involving Stirling numbers|Touchard polynomials}} |
{{Otheruses4|polynomials in variables and exponential functions|the polynomials involving Stirling numbers|Touchard polynomials}} |
||
In [[mathematics]], '''exponential polynomials''' are functions on [[Field (mathematics)|fields]], [[Ring (mathematics)|rings]], or [[abelian group]]s that take the form of [[polynomial]]s in a variable and an exponential function. |
In [[mathematics]], '''exponential polynomials''' are functions on [[Field (mathematics)|fields]], [[Ring (mathematics)|rings]], or [[abelian group]]s that take the form of [[polynomial]]s in a variable and an [[exponential function]]. |
||
==Definition== |
==Definition== |
Revision as of 17:35, 29 April 2009
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
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 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.[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 ex above.[3] 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 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.[4][5]
Uses
Exponential polynomials on R and C often appear in transcendence 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 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.
Notes
- ^ C. J. Moreno, The zeros of exponential polynomials, Compositio Math. 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
- ^ László Székelyhidi, On the extension ofexponential 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.