 Solution

Fix the measure conventions before evaluating this zero-dimensional supersymmetric field theory. Take $dzd\bar{z}$ to mean ordinary positive area measure $d^{2}z=dxdy$, and choose Berezin integration orientations with $\int d^2\theta {\theta }_1{\theta }_2=\int d^2\overline{\theta }{\overline{\theta }}_1{\overline{\theta }}_2=1$. A different overall measure normalization multiplies all the unnormalized answers by the same constant. Assume $r=\mathrm{deg}P=\mathrm{deg}W-1\ge 1$, so $|P(z)|\to \infty$ at infinity and all polynomial insertions converge. An affine or constant $W$ does not give this confining integral and is outside this convergence hypothesis.

The Grassmann variables truncate the fermionic exponential. The coefficient of ${\theta }_1{\theta }_2{\overline{\theta }}_1{\overline{\theta }}_2$ is $P^{\prime }(z)\overline{P^{\prime }(z)}$, so the partition function becomesThe Jacobian determinant of the holomorphic map $w=P(z)$ is $|P^{\prime }(z){|}^2$. Away from its finitely many critical values, a polynomial of polynomial degree $r$ has $r$ preimages counted with multiplicity. The change of variables formula therefore givesCritical values form a set of area zero and do not affect the integral. This partition function measures the covering degree, even when some zeros of $P$ are repeated.