  Tutte initiated the study of nowhere-zero flows and proved the following
fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for
every abelian group $\Gamma$ of order $n$, the number of nowhere-zero
$\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected
orientations), the situation is more subtle. For a finite group $\Gamma$, let
$\epsilon_2(\Gamma)$ be the largest integer $d$ so that $\Gamma$ has a subgroup
isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d
\ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of
nowhere-zero $\Gamma$-flows in $G$ for every abelian group $\Gamma$ with
$\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Beck and Zaslavsky had
previously established the special case of this result when $d=0$ (i.e., when
$\Gamma$ has odd order).
