addition-theorem polynomials Suppose a function f(u) identically satisfies an equation of the form G{f(u+v),f(u),f(v)}=0 for all u and v and u+v in its domain.  Here G(Z,X,Y) is a non vanishing polynomial in the three variables with constant coefficients.  Then one says that f admits an ALGEBRAIC ADDITION THEOREM.  IF f(u) is cos(u), then 
$G(Z,X,Y)=Z^2-2XYZ+X^2+Y^2-1,$
while, if f(u) is the Weierstrass p-function with invariants g_2 and g_3, then 
$G(Z,X,Y)=16(X+Y+Z)^2(X-Y)^2 -8(X+Y+Z){4(X^3+Y^3)-g_2(X+Y)-2g_3}
           +4(X^2+4XY+4Y^2-g_2)^2$
Here is the question: Characterize those polynomials G(Z,X,Y) which express an algebraic addition theorem.
 A: Here is a very basic comment no one has made yet: If $f(u)$ is a rational function of $u$, then there will be some nonzero polynomial $G$ such that $G(f(u), f(v), f(u+v))=0$. That's because $\mathbb{C}(u, v, u+v)$ has transcendence degree $2$ over $\mathbb{C}$.
The same argument applies if $f$ is a rational function of $e^u$, or if $f$ is a rational function of $\wp(u)$ and $\wp'(u)$, where $\wp$ is the Weierstrass $\wp$-function.
Can we show that every example is of one of these forms?
A: It is a famous theorem of Weierstrass that the only meromorphic functions admitting an algebraic addition theorem are rational functions, or rational functions of the exponential function, or elliptic functions.  What has NOT been answered is:  given a polynomial G(Z,X,Y), in the three variables X,Y,Z, is it an addition-theorem polynomial?  Which formal characteristics of G characterize it as such a polynomial?  As far as I know, this question has never been investigated.
A: The examples listed in David Speyer's answer are all of them. This is equivalent to say that all one dimensional algebraic groups are isomorphic to the additive group, the multiplicative group or an elliptic curve. A proof in the language of "algebraic addition theorems" is given in the old book of H. Hancock, Lectures on the theory of elliptic functions, Ch. XXI. 
http://books.google.com/books?id=GDYNAAAAYAAJ&printsec=frontcover&dq=Hancock+elliptic+functions&cd=1#v=onepage&q=&f=false
A: If X' means the derivative with respect to u and Y' that wrsp y, etc., then one condition is: Elimination of Z between G=0 and $X'\frac{\partial G}{\partial Y}=Y'\frac{\partial G}{\partial X}$ leads to only a single equation between X and X' for all values of Y and Y' (see Forsyth, page 357) – Mark B Villarino 0 secs ago 
