be expressed as a polynomial function of the action.

The **Zoll polynomial** $P:{{\mathbb R}}\to{{\mathbb R}}$ associated with $(\mathfrak p_0,c_0)$ is given by $$P(0)=0,\qquad \frac{{{\mathrm d}}P}{{{\mathrm d}}A}(A)=\langle (Ae_0+c_0)^{n},[M_0]\rangle.$$ For instance, when $n=1$, the polynomial reads $$P(A)=\frac12 \langle e_0,[M_0]\rangle A^2+\langle c_0,[M_0]\rangle A.$$

\[t:oszoll\] There is a well-defined volume function $${{\mathfrak{Vol}}}:\mathfrak Z_C(\Sigma)\to{{\mathbb R}},\qquad {{\mathfrak{Vol}}}(\mathfrak p,c):={{\mathfrak{Vol}}}(\mathfrak p^*{\mathrm{\omega}})$$ and well-defined action functional $$\mathcal A:\mathfrak Z_C^0(\Sigma)\to{{\mathbb R}},\qquad \mathcal A(\mathfrak p,c):=\widetilde{\mathcal A}_{\mathfrak p^*{\mathrm{\omega}}}\big([\delta_r]\big).$$ Here, ${\mathrm{\omega}}$ is any closed two-form on $M$ in the class $c$, $\{\delta_r\}$ is any path of periodic curves such that $\delta_r$ is an oriented $\mathfrak p_r$-fibre, where $\{\mathfrak p_r\}$ is any path of oriented $S^1$-bundles from $\mathfrak p_0$ to $\mathfrak p_1=\mathfrak p$. Moreover, there holds $$P\big(\mathcal A(\mathfrak p,c)\big)={{\mathfrak{Vol}}}(\mathfrak p,c),\qquad\forall\,(\mathfrak p,c)\in\mathfrak Z_C^0(\Sigma).$$ If $A_*:=\mathcal A(\mathfrak p_{\Omega_*},c_*)$, then $\tfrac{{{\mathrm d}}P}{{{\mathrm d}}A}(A_*)>0$. In particular, the polynomial $P$ is non-zero.

From this result, we can generalise the equality cases in Theorem \[thm:contact\_sys\] and Proposition \[p:hamsys\].

\[c:zollequality\] Let $\Omega\in{{\mathcal Z}}_C(\Sigma)$ be a Zoll odd-symplectic form such that $\mathfrak p_\Omega\in\mathfrak P^0(\Sigma)$. If we set $\mathcal A(\Omega):=\mathcal A(\mathfrak p_\Omega,[{\mathrm{\omega}}_\Omega])$, then $$P(\mathcal A(\Omega))={{\mathfrak{Vol}}}(\Omega).\eqno\qed$$

In what follows, we describe a conjectural systolic-diastolic inequality for odd-symplectic forms close to $\Omega_*$ and with class $C\in H^2_{{{\mathrm{dR}}}}(\Sigma)$. To this end, we fix a finite open 