= 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.
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