Algebraic addition theorem

A meromorphic function $f$ has an algebraic addition theorem if some nonzero polynomial $P\in \mathbb{C}[U,V,W]$ satisfies $P(f(z),f(w),f(z+w))=0$ wherever the three values are finite. For a polynomial $f$ of degree $d>0$, take roots $z_i$ of $f(z)-U$ and $w_j$ of $f(w)-V$; the product ${\prod }_{i,j}(W-f(z_i+w_j))$ has polynomial coefficients in $U,V$ by the Fundamental theorem of symmetric polynomials, is monic in $W$, and supplies an addition relation. An elliptic function also has such a relation, by finite algebraic extension of its function field and the Weierstrass elliptic function addition formula.

The displayed entire function has exactly the periods $2\pi i\mathbb{Z}$: a period $c$ would make $e^z(e^c-1)$ a constant integer multiple of $2\pi i$, forcing $e^c=1$. If a polynomial addition relation existed, specialize its middle argument at a point with $e^c=\alpha >0$ irrational, avoiding the finitely many specializations making the entire polynomial vanish. With $t=e^z$, the relation becomes a nonzero finite sum of exponentials $e^{(m+\alpha n)t}$. These real exponents are distinct, so the largest one dominates as $t\to +\infty$, a contradiction. This demonstrates that an algebraic addition theorem is stronger than simple periodicity.