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Set $P(z)=W^{\prime }(z)$ and $\overline{P}(\bar{z})=\overline{W^{\prime }(z)}$, treating $z,\bar{z}$ as independent coordinates for Wirtinger derivatives. Use left Grassmann derivatives, so ${\partial }_{{\theta }_2}({\theta }_1{\theta }_2)=-{\theta }_1$. The given odd vector field acts by $V_{1}z={\theta }_1$, $V_1{\theta }_2=-\overline{P}$, and zero on ${\theta }_1,\bar{z},{\overline{\theta }}_i$. Consequently its square vanishes on every coordinate. The square of an odd graded derivation is an even graded derivation, so the nilpotent operator property holds on every function:

For the action in this zero-dimensional supersymmetric field theory, ${\partial }_{z}S=P^{\prime }\overline{P}-P^{\prime \prime }{\theta }_1{\theta }_2$ and ${\partial }_{{\theta }_2}S=P^{\prime }{\theta }_1$. The cubic term vanishes after multiplication by ${\theta }_1$, and the other two terms cancel:Three further independent odd symmetries areFor $V_2$, use ${\partial }_{{\theta }_1}S=-P^{\prime }{\theta }_2$; the same cancellation proves invariance. The barred calculations exchange barred and unbarred variables and coefficients. Each graded derivation is also a nilpotent operator, by its coordinate action. Their different odd coefficients and coordinate derivatives make them linearly independent. These are odd symmetries of a zero-dimensional polynomial model.

The odd symmetries of a zero-dimensional polynomial model preserve both the action and the flat integration measure: their coefficients have zero superdivergence. The confining polynomial weight removes boundary terms at infinity. Thus the integral of an odd-symmetry derivative is zero, a supersymmetric Ward identity for this finite-dimensional integral in this zero-dimensional supersymmetric field theory.

Since $g(z)$ is a holomorphic function, the barred symmetry obeys ${\overline{V}}_1({\overline{\theta }}_{2}g(z))=-P(z)g(z)$. Applying the supersymmetric Ward identity gives the requested unnormalized insertion:One can also verify the result after Berezin integration, without invoking the symmetry terminology:The final integration by parts uses ${\partial }_{\bar{z}}(g{P}^{\prime })=0$ and rapid decay. The derivative on $W^{\prime }$ in the insertion is essential.

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.

If an odd graded derivation $Q$ preserves the action and integration measure, then $⟨QX⟩=0$ whenever boundary contributions and anomalies vanish. This applies to finite-dimensional zero-dimensional supersymmetric field theories as well as suitably regulated field integrals.