In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an addition theorem is a formula such as that for the
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 ...
:
:''e''
''x'' + ''y'' = ''e''
''x'' · ''e''
''y'',
that expresses, for a particular
function ''f'', ''f''(''x'' + ''y'') in terms of ''f''(''x'') and ''f''(''y''). Slightly more generally, as is the case with the
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and , several functions may be involved; this is more apparent than real, in that case, since there is an
algebraic function of (in other words, we usually take their functions both as defined on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
).
The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s. To "classify" addition theorems it is necessary to put some restriction on the type of function ''G'' admitted, such that
:''F''(''x'' + ''y'') = ''G''(''F''(''x''), ''F''(''y'')).
In this identity one can assume that ''F'' and ''G'' are vector-valued (have several components). An algebraic addition theorem is one in which ''G'' can be taken to be a vector of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, in some set of variables. The conclusion of the mathematicians of the time was that the theory of
abelian function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s essentially exhausted the interesting possibilities: considered as a
functional equation to be solved with polynomials, or indeed
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s or
algebraic functions, there were no further types of solution.
In more contemporary language this appears as part of the theory of
algebraic groups, dealing with
commutative groups. The connected,
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
by rather weak conditions on its group law. The so-called
quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of
formal groups.
See also
*
Timeline of abelian varieties
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
Early history
* c. 1000 Al-Karaji writes on congruent numbers
Seventeenth century
* Fermat studies descent for elliptic curves
* 1643 Ferm ...
*
Addition theorem for spherical harmonics
*
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elli ...
References
*
{{DEFAULTSORT:Addition Theorem
Theorems in algebraic geometry
Theorems in algebra