The homogeneity and pole-order conjecture for the pairing on the Gauss–Manin system

Let $f$ be the linear free divisor considered in the paper, let $\mathbf{G}$ and $\overline{\mathbf{G}}$ be the corresponding Gauss–Manin systems, and let $G$ and $\overline{G}$ denote their lattices. Let $\tau$ and $t$ be the parameters, with grading conventions $\deg(\tau)=-1$ and $\deg(t)=n$. The pairing $S$ from the canonical solution theorem is conjectured to extend to the localization in $t$ and to be meromorphic at $t=0$, inducing a pairing

$$S:\mathbf{G}[t^{-1}]\otimes\overline{\mathbf{G}}[t^{-1}]\longrightarrow \mathbb{C}[\tau,\tau^{-1},t,t^{-1}].$$

Moreover, with the natural grading induced from the grading of $\Omega^{n-1}(\log h)$, Homogeneity and pole-order conjecture. The pairing is homogeneous, in the sense that

$$S\bigl((\mathbf{G}[t^{-1}])_k\otimes(\overline{\mathbf{G}}[t^{-1}])_l\bigr)\subseteq \mathbb{C}[\tau,\tau^{-1},t,t^{-1}]_{k+l},$$

and it satisfies

$$S(G\otimes\overline{G})\subseteq \tau^{-n+1}\mathbb{C}[\tau^{-1},t].$$

This additional control of the pairing is intended to make it possible to specialize the Frobenius structures constructed for nonzero parameters to the divisor at $t=0$. The parser supplies no evidence that the conjecture has been resolved.