Computation

By the finite determinacy theorem, we may assume that $f\in\mathbf{C}[\boldsymbol{x}]$, $\boldsymbol{x}:=x_0,\dots,x_n$, is a polynomial. Since $\mathbf{C}[\boldsymbol{x}]_{(\boldsymbol{x})}\subset\mathbf{C}\{\boldsymbol{x}\}$ is faithfully flat and all data will be defined over $\mathbf{C}[\boldsymbol{x}]_{(\boldsymbol{x})}$, we may replace $\mathbf{C}\{\boldsymbol{x}\}$ by $\mathbf{C}[\boldsymbol{x}]_{(\boldsymbol{x})}$ and, similarly, ${\mathbf{C}\{t\}}$ by $\mathbf{C}[t]_{(t)}$ and ${\mathbf{C}\{\!\{s\}\!\}}$ by $\mathbf{C}[s]_{(s)}$ for the computation. With the additional assumption $f\in\mathbf{Q}[\boldsymbol{x}]$, all data will be defined over $\mathbf{Q}$, and we can apply methods of computer algebra. Using standard basis methods for local rings, $\mathbf{C}[\boldsymbol{x}]_{\langle \boldsymbol{x}\rangle}$ one can compute a monomial $\mathbf{C}$-basis $\boldsymbol{m}=(m_1,\dots,m_\mu)^T$ of

$$\Omega_f:=\Omega^{n+1}/{d\!\,} f\wedge\Omega^n\cong{\mathbf{C}\{\boldsymbol{x}\}}/(\partial_{\boldsymbol{x}} f).$$

Since $\mathcal{H}''/s\mathcal{H}''\cong\Omega_f$, $\boldsymbol{m}$ represents a ${\mathbf{C}\{\!\{s\}\!\}}$-basis of $\mathcal{H}''$ and a $\mathbf{C}(s)$-basis of $\mathcal{G}$ by Nakayama's lemma.

The matrix $A$ of $t$ with respect to $\boldsymbol{m}$ is defined by $t\boldsymbol{m}=:A\boldsymbol{m}$. Since $t=s^2\partial_s$, we obtain for $g\in\mathbf{C}(s)^\mu$

$$tg\boldsymbol{m}\,=\,\bigl(gA+s^2\partial_s(g)\bigr)\boldsymbol{m}\,.$$

So the action of $t$ in terms of the $\mathbf{C}(s)$-basis $\boldsymbol{m}$ is determined by the matrix $A$ by the above formula.

A reduced normalform with respect to a local monomial ordering allows to compute the projection to the first summand in

$$\SelectTips{cm}{}\xymatrix {\Omega_f\oplus{d\!\,} f\wedge\Omega^n... ...e{d\!\,}\Omega^{n-1}\cong \mathcal{H}''/s\mathcal{H}''\oplus s\mathcal{H}''}.$$

Since $t[\omega]=[f\omega]$ and $s^{-1}[{d\!\,} f\wedge\eta]=\partial_t[{d\!\,} f\wedge\eta]=[{d\!\,}\eta]$, the matrix $A$ of $t$ with respect to $\boldsymbol{m}$ can be computed up to arbitrarily high order.

The basis representation $H''_k$ of $\mathcal{H}''_k$ with respect to $\boldsymbol{m}$ defined by $\mathcal{H}''_k=:H''_k\boldsymbol{m}$ can be computed inductively by

$${\delta\!H}_0 :=H_0:={\mathbf{C}\{\!\{s\}\!\}}^\mu,$$

$${\delta\!H}_{k+1} :=jet_{-1}\bigl(s^{-1}{\delta\!H}_kjet_k(A)+s\partial_s{\delta\!H}_k\bigr),$$

$$H_{k+1} =H_k+{\delta\!H}_{k+1}.$$

Using standard basis methods, one can check if $H_k=H_{k+1}$ and compute a ${\mathbf{C}\{\!\{s\}\!\}}$-basis ${\boldsymbol{m}}'$ of $H_{k_\infty}=:H_\infty$ with

$$\delta({\boldsymbol{m}}'):=\max\bigl\{ord\bigl(m'_{i_1,j_1}\bigr)... ...,j_2}\bigr)\:\big\vert\: m'_{i_1,j_1}\ne0\ne m'_{i_2,j_2}\bigr\}\le k_\infty.$$

Then the matrix $A'$ of $t$ with respect to the ${\mathbf{C}\{\!\{s\}\!\}}$-basis ${\boldsymbol{m}}'\boldsymbol{m}$ of $\mathcal{H}''_\infty$ is defined by the formula $\boldsymbol{m}'A+s^2\partial_s \boldsymbol{m}'=:A'\boldsymbol{m}'$, and $jet_k(A')=jet_k(A'_{\le k})$ for $A'_{\le k}$ defined by

$$\boldsymbol{m}'jet_{k+\delta(\boldsymbol{m}')}(A)+s^2\partial_s \boldsymbol{m}'=:A'_{\le k}\boldsymbol{m}'.$$

Hence, the basis representation of $\overline{\partial_t t}\in End_\mathbf{C}\bigl(\mathcal{H}''_\infty\big/ \partial_t^{-K}\mathcal{H}''_\infty\bigr)$

with respect to $\boldsymbol{m}'\boldsymbol{m}$ is

$$\overline{s^{-1}A'+s\partial_s}\,=\, \overline{s^{-1}A'_{\le K}+s... ...{C}\bigl({\mathbf{C}\{\!\{s\}\!\}}^\mu/s^K{\mathbf{C}\{\!\{s\}\!\}}^\mu\bigr).$$

The basis representation $H'$ of $\mathcal{H}''$ with respect to $\boldsymbol{m}'\boldsymbol{m}$ is defined by $H_0=:H'\boldsymbol{m}'$, and $V^{\bullet+1}_{\overline{s^{-1}A'_{\le K}+s\partial_s}}(H'/sH')\boldsymbol{m}'$ is the basis representation of $V(\mathcal{H}''/s\mathcal{H}'')$ with respect to $\boldsymbol{m}$. The matrix of $\overline{s^{-1}A'_{\le K}+s\partial_s}$ with respect to the canonical $\mathbf{C}$-basis

$$\setcounter {MaxMatrixCols}{20}\begin{pmatrix} 1 & & & s & & & s^... ...cdots & & \ddots & \\ & & 1 & & & s & & & s^2 & & & & s^{K-1} \end{pmatrix}^t$$

of ${\mathbf{C}\{\!\{s\}\!\}}^\mu/s^K{\mathbf{C}\{\!\{s\}\!\}}^\mu$ is given by the block matrix

$$\begin{pmatrix} A'_1 & A'_2 & A'_3 & A'_4 & \cdots & A'_K \\ & A... ... \ddots & \vdots \\ & & & & \ddots & A'_2 \\ & & & & & A'_1+K-1 \end{pmatrix}$$

where $A'=\sum_{k\ge0}A'_ks^k$. Since the eigenvalues of $A'_1$ are rational, they can be computed using univariate factorization over the rational numbers. Then the V-filtration $V^{\bullet+1}_{\overline{s^{-1}A'_{\le K}+s\partial_s}}$ can be computed using methods of linear algebra.