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.*