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In mathematics, exponential polynomials are functions on
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
,
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film an ...
, or
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s that take the form of
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s in a variable and an
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
.


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 In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation. Definition A field is an algebraic structure composed of a set of elements, ''F'', two binary operations, ...
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''''x''1, ..., ''e''''x''''n''), where ''P'' is a polynomial in 2''n'' variables. For formal exponential polynomials over a field ''K'' we proceed as follows. Let ''W'' be a finitely generated Z-
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
of ''K'' and consider finite sums of the form :\sum_^ 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
''G'' a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
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''.P. G. Laird, ''On characterizations of exponential polynomials'', Pacific Journal of Mathematics 80 (1979), pp.503–507.


Properties

Ritt's theorem states that the analogues of
unique factorization In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
and the
factor theorem In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial f(x) has a factor (x - \alpha) if and only if f(\alpha)= ...
hold for the ring of exponential polynomials.


Applications

Exponential polynomials on R and C often appear in
transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendenc ...
, where they appear as auxiliary functions in proofs involving the exponential function. They also act as a link between
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the st ...
and
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineer ...
. 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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
and
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 terms of model theory, Wilkie's theo ...
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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
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 In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every
over R. Exponential polynomials appear in the characteristic equation associated with linear delay differential equations.


Notes

{{Reflist Polynomials