2·1. The equivariant Grothendieck ring of varieties

Suppose X is a separated finite type scheme over k. We let $K_0(\textbf{Var}_X)$ denote the Grothendieck ring of varieties over X, we let $\mathbb{L} \in K_0(\textbf{Var}_X)$ denote the class of $\mathbb{A}_k^1 \times_k X$ , and for each separated finite type X-scheme Y, we let $[Y/X] \in K_0(\textbf{Var}_X)$ denote the class of Y. We will let $\mathscr{M}_X$ denote the ring obtained by inverting $\mathbb{L}$ in $K_0(\textbf{Var}_X)$ , and by slight abuse of notation, we write $\mathbb{L}, [Y/X] \in \mathscr{M}_X$ to denote the images of $\mathbb{L}, [Y/X]$ , respectively, in $\mathscr{M}_X$ .

We will let $K_0(\textbf{Var}_k)$ and $\mathscr{M}_k$ denote $K_0(\textbf{Var}_{\text{Spec}(k)})$ and $\mathscr{M}_{\text{Spec}(k)}$ , respectively, and for each separated finite type k-scheme Y, we will write $[Y] = [Y/\text{Spec}(k)]$ in both $K_0(\textbf{Var}_k)$ and $\mathscr{M}_k$ .

Suppose G is a finite abelian group. An action of G on a scheme is said to be good if each orbit is contained in an affine open subscheme. For example, any G-action on any quasiprojective k-scheme is good. If the separated finite type k-scheme X is endowed with a good G-action, then we will let $K_0^G(\textbf{Var}_X)$ denote the G-equivariant Grothendieck ring of varieties over X. For the precise definition of $K_0^G(\textbf{Var}_X)$ , we refer to [ Reference HartmannHar15 , definition 4·1]. We will let $\mathbb{L} \in K_0^G(\textbf{Var}_X)$ denote the class of $\mathbb{A}_k^1 \times_k X$ with the action induced by the trivial G-action on $\mathbb{A}_k^1$ and the given G-action on X, and for each separated finite type X-scheme Y with good G-action making the structure morphism G-equivariant, we will let $[Y/X,G] \in K_0^G(\textbf{Var}_X)$ denote the class of Y with its given G-action. We will let $\mathscr{M}_X^G$ denote the ring obtained by inverting $\mathbb{L}$ in $K_0^G(\textbf{Var}_X)$ , and by slight abuse of notation, we will let $\mathbb{L}, [Y/X,G] \in \mathscr{M}_X^G$ denote the images of $\mathbb{L}, [Y/X, G]$ , respectively, in $\mathscr{M}_X^G$ .

If X is a separated finite type k-scheme with no specified G-action and we refer to $K_0^G(\textbf{Var}_X)$ or $\mathscr{M}_X^G$ , then we are considering X with the trivial G-action. We will let $K_0^G(\textbf{Var}_k)$ and $\mathscr{M}_k^G$ denote $K_0^G(\textbf{Var}_{\text{Spec}(k)})$ and $\mathscr{M}_{\text{Spec}(k)}^G$ , respectively, and for each separated finite type k-scheme Y with good G-action making the structure morphism G-equivariant, we will let $[Y, G] = [Y/\text{Spec}(k), G]$ in both $K_0^G(\textbf{Var}_k)$ and $\mathscr{M}_k$ .

For $\ell \in \mathbb{Z}_{>0}$ , we will let $\mu_\ell \subset k^\times$ denote the group of $\ell$ th roots of unity.

For each $\ell, m \in \mathbb{Z}_{>0}$ , there is a morphism $\mu_{\ell m} \to \mu_\ell\,:\, \xi \mapsto \xi^m$ . Suppose that X is a separated finite type scheme over k. Then the morphism $\mu_{\ell m} \to \mu_\ell$ induces ring morphisms $K_0^{\mu_\ell}(\textbf{Var}_X) \to K_0^{\mu_{\ell m}}(\textbf{Var}_X)$ and $\mathscr{M}_X^{\mu_\ell} \to \mathscr{M}_X^{\mu_{\ell m}}$ . We will let $K_0^{\hat\mu}(\textbf{Var}_X) = \varinjlim_{\ell}K_0^{\mu_\ell}(\textbf{Var}_X)$ and $\mathscr{M}_X^{\hat\mu} = \varinjlim_\ell \mathscr{M}_X^{\mu_\ell}$ . We let $\mathbb{L} \in K_0^{\hat\mu}(\textbf{Var}_X)$ denote the image of $\mathbb{L} \in K_0^{\mu_\ell}(\textbf{Var}_X)$ for any $\ell \in \mathbb{Z}_{>0}$ , and similarly we let $\mathbb{L} \in \mathscr{M}_X^{\hat\mu}$ denote the image of $\mathbb{L} \in \mathscr{M}_X^{\mu_\ell}$ for any $\ell \in \mathbb{Z}_{>0}$ . For each separated finite type X-scheme Y with good $\mu_\ell$ -action making the structure morphism $\mu_\ell$ -equivariant, we let $[Y/X, \hat\mu] \in K_0^{\hat\mu}(\textbf{Var}_X)$ denote the image of $[Y/X, \mu_\ell] \in K_0^{\mu_\ell}(\textbf{Var}_X)$ , and we similarly let $[Y/X, \hat\mu] \in \mathscr{M}_X^{\hat\mu}$ denote the image of $[Y/X, \mu_\ell] \in \mathscr{M}_X^{\mu_\ell}$ .

We will let $K_0^{\hat\mu}(\textbf{Var}_k)$ and $\mathscr{M}_k^{\hat\mu}$ denote $K_0^{\hat\mu}(\textbf{Var}_{\text{Spec}(k)})$ and $\mathscr{M}_{\text{Spec}(k)}^{\hat\mu}$ , respectively, and for each $\ell \in \mathbb{Z}_{>0}$ and each separated finite type k-scheme Y with good $\mu_\ell$ -action making the structure morphism $\mu_\ell$ -equivaraint, we will let $[Y, \hat\mu] = [Y/\text{Spec}(k), \hat\mu]$ in both $K_0^{\hat\mu}(\textbf{Var}_k)$ and $\mathscr{M}_k^{\hat\mu}$ .

2·2. The motivic zeta functions of Denef and Loeser

Let X be a smooth, pure dimensional, separated, finite type k-scheme. For each $\ell \in \mathbb{Z}_{\geq 0}$ , we will let $\mathscr{L}_\ell(X)$ denote $\ell$ -th jet scheme of X, and for each $m \geq \ell$ , we will let $\theta^m_\ell\,:\, \mathscr{L}_m(X) \to \mathscr{L}_\ell(X)$ denote the truncation morphism. We will let $\mathscr{L}(X) = \varprojlim_\ell \mathscr{L}_\ell(X)$ denote the arc scheme of X, and for each $\ell \in \mathbb{Z}_{\geq 0}$ , we will let $\theta_\ell\,:\, \mathscr{L}(X) \to \mathscr{L}_\ell(X)$ denote the canonical morphism. The following is a special case of a theorem of Bhatt’s [ Reference BhattBha16 , theorem 1·1].

A subset of $\mathscr{L}(X)$ is called a cylinder if it is the preimage, under $\theta_\ell$ , of a constructible subset of $\mathscr{L}_\ell(X)$ for some $\ell \in \mathbb{Z}_{\geq 0}$ . We will let $\mu_X$ denote the motivic measure on $\mathscr{L}(X)$ , which assigns a motivic volume in $\mathscr{M}_X$ to each cylinder.

Suppose f is a regular function on X. If $x \in \mathscr{L}(X)$ has residue field k(x), then it corresponds to a k-morphism $\psi_x\,:\, \text{Spec}(k(x)[\![ \pi ]\!]) \to X$ , and we will let f(x) denote $f(\psi_x) \in k(x)[\![ \pi ]\!]$ . For each $x \in \mathscr{L}(X)$ , the order of f at x will refer to the order of $\pi$ in the power series f(x), and the angular component of f at x will refer to the leading coefficient of the power series f(x). We will let $\text{ord}_f\,:\, \mathscr{L}(X) \to \mathbb{Z}_{\geq 0} \cup \{\infty\}$ denote the function taking each $x \in \mathscr{L}(X)$ to the order of f at x. We will let $Z^{\textrm{naive}}_f(T) \in \mathscr{M}_X[\![ T]\!]$ denote the motivic Igusa zeta function of f. Then

\[Z^{\textrm{naive}}_f(T) = \sum_{\ell \in \mathbb{Z}_{\geq 0}} \mu_X\!\left(\text{ord}_f^{-1}(\ell)\right) T^\ell \in \mathscr{M}_X [\![ T ]\!].\]

We will let $Z_f^{\hat\mu}(T) \in \mathscr{M}_X^{\hat\mu}[\![ T]\!]$ denote the Denef–Loeser motivic zeta function of f. We briefly recall the definition of $Z_f^{\hat\mu}(T)$ . The constant term of $Z_f^{\hat\mu}(T)$ is equal to 0. Let $\ell \in \mathbb{Z}_{>0}$ , and let $Y_{\ell,1}$ be the closed subscheme of $\mathscr{L}_\ell(X)$ where f is equal to $\pi^\ell$ . For any k-algebra A, there is a $\mu_\ell$ -action on $A[\![ \pi ]\!]$ , where $\xi \in \mu_\ell$ acts by $\pi \mapsto \xi \pi$ , and these actions induce a $\mu_\ell$ -action on $\mathscr{L}_\ell(X)$ making $Y_{\ell, 1}$ invariant. Note also that the truncation morphism $\theta^\ell_0\,:\, \mathscr{L}_\ell(X) \to X$ restricts to a $\mu_\ell$ -equivariant morphism $Y_{\ell, 1} \to X$ . Then the coefficient of $T^\ell$ in $Z_f^{\hat\mu}(T)$ is defined to be equal to $[Y_{\ell, 1}/X, \hat\mu]\mathbb{L}^{-(\ell+1)\dim X} \in \mathscr{M}_X^{\hat\mu}$ .

2·3. Hartmann’s equivariant motivic integration

For the remainder of this paper, let $R = k[\![ \pi ]\!]$ , the ring of power series over k. We will set up some notation and recall facts for Greenberg schemes and Hartmann’s equivariant motivic integration [ Reference HartmannHar15 ], which is an equivariant version of Sebag’s motivic integration for formal schemes [ Reference SebagSeb04 ]. For the non-equivariant version of this theory, we also recommend the book [ Reference Chambert–Loir, Nicaise and SebagCNS18 ].

Let $\mathfrak{X}$ be a smooth, pure relative dimensional, separated, finite type R-scheme. We will let $\mathfrak{X}_0$ denote the special fiber of $\mathfrak{X}$ . For each $\ell \in \mathbb{Z}_{\geq 0}$ , we will let $\mathscr{G}_\ell(\mathfrak{X})$ denote the $\ell$ -th Greenberg scheme of $\mathfrak{X}$ . Thus $\mathscr{G}_\ell(\mathfrak{X})$ represents the functor taking each k-algebra A to $\text{Hom}_R\!\left(\text{Spec}\!\left(A[\pi]/\left(\pi^{\ell+1}\right)\right), \mathfrak{X}\right)$ . For each $m \geq \ell$ , we will let $\theta^m_\ell\,:\, \mathscr{G}_m(\mathfrak{X}) \to \mathscr{G}_\ell(\mathfrak{X})$ denote the truncation morphism. We will let $\mathscr{G}(\mathfrak{X}) = \varprojlim_\ell \mathscr{G}_\ell(\mathfrak{X})$ denote the Greenberg scheme of $\mathfrak{X}$ , and for each $\ell \in \mathbb{Z}_{\geq 0}$ , we will let $\theta_\ell\,:\, \mathscr{G}(\mathfrak{X}) \to \mathscr{G}_\ell(\mathfrak{X})$ denote the canonical morphism. As for arc schemes, the following is a special case of [ Reference BhattBha16 , theorem 1·1]. See for example [ Reference Chambert–Loir, Nicaise and SebagCNS18 , chapter 4, proposition 3·1·7].

A subset of $\mathscr{G}(\mathfrak{X})$ is called a cylinder if it is the preimage, under $\theta_\ell$ , of a constructible subset of $\mathscr{G}_\ell(\mathfrak{X})$ for some $\ell \in \mathbb{Z}_{\geq 0}$ . We will let $\mu_\mathfrak{X}$ denote the motivic measure on $\mathscr{G}(\mathfrak{X})$ , which assigns a motivic volume in $\mathscr{M}_{\mathfrak{X}_0}$ to each cylinder.

Suppose f is a regular function on $\mathfrak{X}$ . If $x \in \mathscr{G}(\mathfrak{X})$ has residue field k(x), then it corresponds to an R-morphism $\psi_x\,:\, \text{Spec}(k(x)[\![ \pi ]\!]) \to \mathfrak{X}$ , and we will let f(x) denote $f(\psi_x) \in k(x)[\![ \pi ]\!]$ . As for arc schemes, this is used to define the order and angular component of f at x and the order function $\text{ord}_f\,:\, \mathscr{G}(\mathfrak{X}) \to \mathbb{Z}_{\geq 0} \cup \{\infty\}$ .

Now suppose G is a finite abelian group acting on R, and suppose that each element of G acts on R by a $\pi$ -adically continuous k-algebra morphism. Endow $\mathfrak{X}$ with a good G-action making the structure morphism G-equivariant, and endow $\mathfrak{X}_0$ with the restriction of the G-action on $\mathfrak{X}$ . The G-action on $\mathfrak{X}$ induces good G-actions on $\mathscr{G}(\mathfrak{X})$ and each $\mathscr{G}_\ell(\mathfrak{X})$ . We refer to [ Reference HartmannHar15 , section 3·2] for the construction and properties of these G-actions on the Greenberg schemes. We will let $\mu_{\mathfrak{X}}^G$ denote the G-equivariant motivic measure on $\mathscr{G}(\mathfrak{X})$ , which assigns a motivic volume in $\mathscr{M}_{\mathfrak{X}_0}^G$ to each G-invariant cylinder in $\mathscr{G}(\mathfrak{X})$ . We refer to [ Reference HartmannHar15 , section 4·2] for the definition of $\mu_{\mathfrak{X}}^G$ .

If $A \subset \mathscr{G}(\mathfrak{X})$ is a G-invariant cylinder and $\alpha\,:\, A \to \mathbb{Z}$ is a function whose fibers are G-invariant cylinders, then the integral of $\alpha$ is defined to be

\[\int_A \mathbb{L}^{-\alpha} \textrm{d}\mu_{\mathfrak{X}}^G = \sum_{\ell \in \mathbb{Z}} \mu_\mathfrak{X}^G\left(\alpha^{-1}(\ell)\right) \mathbb{L}^{-\ell} \in \mathscr{M}_{\mathfrak{X}_0}^G.\]

We now state the equivariant version of the motivic change of variables formula [ Reference HartmannHar15 , theorem 4·18]. If $h\,:\, \mathfrak{Y} \to \mathfrak{X}$ is a morphism of R-schemes, then we let $\text{ordjac}_h\,:\, \mathscr{G}(\mathfrak{Y}) \to \mathbb{Z}_{\geq 0} \cup \{\infty\}$ denote the order function of the jacobian ideal of h.

Theorem 2·8 (Hartmann). Suppose $\# G$ is not divisible by the characteristic of k. Let $\mathfrak{X}, \mathfrak{Y}$ be smooth, pure relative dimensional, separated, finite type R-schemes with good G-action making the structure morphisms equivariant, and let $h\,:\, \mathfrak{Y} \to \mathfrak{X}$ be a G-equivariant morphism that induces an open immersion on generic fibers. Let A,B be G-invariant cylinders in $\mathscr{G}(\mathfrak{X}), \mathscr{G}(\mathfrak{Y})$ , respectively, such that h induces a bijection $B(k^{\prime}) \to A(k^{\prime})$ for all extensions k^′ of k.

If $\alpha\,:\, A \to \mathbb{Z}$ is a function whose fibers are G-invariant cylinders, then $\alpha \circ \mathscr{G}(h) - \text{ordjac}_h\,:\, B \to \mathbb{Z}$ is a function whose fibers are G-invariant cylinders, and

\[\int_A \mathbb{L}^{-\alpha} \textrm{d}\mu_{\mathfrak{X}}^G = \int_B \mathbb{L}^{-\left(\alpha \circ \mathscr{G}(h) + \text{ordjac}_h\right)} \textrm{d}\mu_{\mathfrak{Y}}^G \in \mathscr{M}_{\mathfrak{X}_0}^G.\]

We note that for all $\ell \in \mathbb{Z}_{>0}$ , the characteristic of k never divides $\# \mu_\ell$ .

2·4. Linear subspaces and matroids

Let $d, n \in \mathbb{Z}_{>0}$ . We will let $\textrm{Gr}_{d,n}$ denote the Grassmannian of d-dimensional linear subspaces in $\mathbb{A}_k^n = \text{Spec}(k[x_1, \dots, x_n])$ . We will let $\mathbb{G}_{m,k}^n = \text{Spec}\!\left(k\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]\right) \subset \mathbb{A}_k^n$ denote the complement of the coordinate hyperplanes, and we will let $V(x_1 \cdots x_n -1)$ denote the closed subscheme of $\mathbb{A}_k^n$ defined by $(x_1 \cdots x_n -1)$ . For each $\mathcal{A} \in \textrm{Gr}_{d,n}(k)$ , we will let $X_\mathcal{A} \hookrightarrow \mathbb{A}_k^n$ denote the corresponding linear subspace. If $X_\mathcal{A}$ is not contained in a coordinate hyperplane of $\mathbb{A}_k^n$ , then the restrictions to $X_\mathcal{A}$ of the coordinates $x_i$ define a central essential hyperplane arrangement in $X_\mathcal{A}$ . We let $U_\mathcal{A} = X_\mathcal{A} \cap \mathbb{G}_{m,k}^n$ and $F_\mathcal{A} = X_\mathcal{A} \cap V(x_1 \cdots x_n -1)$ denote this arrangement’s complement and Milnor fiber, respectively, and we endow $F_\mathcal{A}$ with the restriction of the $\mu_n$ -action on $\mathbb{A}_k^n$ where each $\xi \in \mu_n$ acts by scalar multiplication. In the context of tropical geometry, we will consider both $U_\mathcal{A}$ and $F_\mathcal{A}$ as closed subschemes of the algebraic torus $\mathbb{G}_{m,k}^n$ . We will let $Z_{\mathcal{A}}^{\hat\mu}(T) \in \mathscr{M}^{\hat\mu}_{X_\mathcal{A}}[\![ T ]\!]$ and $Z_{\mathcal{A}}^{\textrm{naive}}(T) \in \mathscr{M}_{X_\mathcal{A}} [\![ T ]\!]$ denote the Denef–Loeser motivic zeta function and the motivic Igusa zeta function, respectively, of the restriction of the monomial $x_1 \cdots x_n$ to $X_\mathcal{A}$ . We will let $Z^{\hat\mu}_{\mathcal{A}, k}(T) \in \mathscr{M}_k^{\hat\mu} [\![ T ]\!]$ $\big($ resp. $Z^{\textrm{naive}}_{\mathcal{A}, k}(T) \in \mathscr{M}_k [\![ T ]\!]\big)$ denote the power series obtained by pushing forward each coefficient of $Z_{\mathcal{A}}^{\hat\mu}(T)$ $\big($ resp. $Z_{\mathcal{A}}^{\textrm{naive}}(T)\big)$ along the structure morphism of $X_\mathcal{A}$ . We will let $Z^{\hat\mu}_{\mathcal{A}, 0}(T) \in \mathscr{M}_k^{\hat\mu} [\![ T ]\!]$ $\big($ resp. $Z^{\textrm{naive}}_{\mathcal{A}, 0}(T) \in \mathscr{M}_k [\![ T ]\!]\big)$ denote the power series obtained by pulling back each coefficient of $Z_{\mathcal{A}}^{\hat\mu}(T)$ $\big($ resp. $Z_{\mathcal{A}}^{\textrm{naive}}(T)\big)$ along the inclusion of the origin into $X_\mathcal{A}$ .

Let $\mathcal{M}$ be a rank d loop-free matroid on $\{1, \dots, n\}$ . We will let $\chi_\mathcal{M}(\mathbb{L}) \in \mathscr{M}_k$ denote the characteristic polynomial of $\mathcal{M}$ evaluated at $\mathbb{L}$ , so

\[\chi_\mathcal{M}(\mathbb{L}) = \sum_{I \subset \{1, \dots, n\}} ({-}1)^{\# I} \mathbb{L}^{d-\text{rk}\, I} \in \mathscr{M}_k,\]

where $\text{rk}\, I$ is the rank function of $\mathcal{M}$ applied to I. We will let $\mathcal{B}(\mathcal{M})$ denote the set of bases of $\mathcal{M}$ , and we will let $\text{wt}_\mathcal{M}\,:\, \mathbb{R}^n \to \mathbb{R}$ denote the function $(w_1, \dots, w_n) \mapsto \max_{B \in \mathcal{B}(\mathcal{M})} \sum_{i \in B} w_i$ . For each $w = (w_1, \dots, w_n) \in \mathbb{R}^n$ , we let

\[\mathcal{B}(\mathcal{M}_w) = \left\{B \in \mathcal{B}(\mathcal{M}) \, \biggm\vert \, \sum_{i \in B} w_i = \text{wt}_\mathcal{M}(w)\right\}\]

be the set of all bases with maximal w-weight. The set $\mathcal{B}(\mathcal{M}_w)$ is the set of the bases of a rank d matroid on $\{1, \dots, n\}$ , and we will let $\mathcal{M}_w$ denote that matroid. We let $\text{Trop}(\mathcal{M})$ denote the Bergman fan of $\mathcal{M}$ , so

\[\text{Trop}(\mathcal{M}) = \{w \in \mathbb{R}^n \mid \text{$\mathcal{M}_w$ is loop-free}\}.\]

We will let $\textrm{Gr}_\mathcal{M} \subset \textrm{Gr}_{d,n}$ denote the locus parametrising linear subspaces whose associated hyperplane arrangements have combinatorial type $\mathcal{M}$ . For all $\mathcal{A} \in \textrm{Gr}_\mathcal{M}(k)$ , the fact that $\mathcal{M}$ is loop-free implies that $X_\mathcal{A}$ is not contained in a coordinate hyperplane. Note that if $\mathcal{A} \in \textrm{Gr}_\mathcal{M}(k)$ , then $[U_\mathcal{A}] = \chi_\mathcal{M}(\mathbb{L}) \in \mathscr{M}_k$ and

\[\text{Trop}(U_\mathcal{A}) = \{w \in \mathbb{R}^n \mid \text{in}_w\ U_\mathcal{A} \neq \emptyset\} = \text{Trop}(\mathcal{M}).\]

For each $\mathcal{A} \in \textrm{Gr}_\mathcal{M}(k)$ and each $w \in \text{Trop}(\mathcal{M})$ , we will let $\mathcal{A}_w \in \textrm{Gr}_{\mathcal{M}_w}(k)$ denote the unique point such that $\text{in}_w\ U_\mathcal{A} = U_{\mathcal{A}_w}$ .

Before concluding the preliminaries, we recall two propositions proved in [ Reference Kutler and UsatineKU22 ] that will be used in Section 6. If $B \in \mathcal{B}(\mathcal{M})$ and $i \in \{1, \dots, n\} \setminus B$ , then we will let $C(\mathcal{M}, i, B)$ denote the fundamental circuit in $\mathcal{M}$ of B with respect to i, so $C(\mathcal{M}, i, B)$ is the unique circuit in $\mathcal{M}$ contained in $B \cup \{i\}$ . For each circuit C in $\mathcal{M}$ and each $\mathcal{A} \in \textrm{Gr}_\mathcal{M}(k)$ , we will let $L_C^\mathcal{A} \in k[x_1, \dots, x_n]$ denote a linear form in the ideal defining $X_\mathcal{A}$ in $\mathbb{A}_k^n$ such that the coefficient of $x_i$ in $L_C^\mathcal{A}$ is nonzero if and only if $i \in C$ . Such an $L_C^\mathcal{A}$ exists and is unique up to scaling by a unit in k. Once and for all, we fix such an $L_C^\mathcal{A}$ for all C and $\mathcal{A}$ .

Proposition 2·10 ([ Reference Kutler and UsatineKU22 , proposition 3·5]). Let $\mathcal{A} \in \textrm{Gr}_\mathcal{M}(k)$ , let $w \in \mathbb{R}^n$ , and let $B \in \mathcal{B}(\mathcal{M}_w)$ . Then

\[\left\{L_{C(\mathcal{M}, i,B)}^\mathcal{A} \mid i \in \{1, \dots, n\} \setminus B\right\} \subset k\!\left[x_1, \dots, x_n\right]\]

generates the ideal of $X_\mathcal{A}$ in $\mathbb{A}_k^n$ , and

\[\left\{ \text{in}_w\ L_{C(\mathcal{M}, i, B)}^\mathcal{A} \mid i \in \{1, \dots, n\} \setminus B\right\} \subset k\!\left[x_1^{\pm}, \dots, x_n^{\pm}\right]\]

generates the ideal of $\text{in}_w\ U_{\mathcal{A}}$ in $\mathbb{G}_{m,k}^n$ .

Proposition 2·11 ([ Reference Kutler and UsatineKU22 , proposition 3·2]). Let $w = (w_1, \dots, w_n) \in \mathbb{R}^n$ , let $B \in \mathcal{B}(\mathcal{M}_w)$ , and let $i \in \{1, \dots, n\} \setminus B$ . Then

\[\min_{j \in C(\mathcal{M},i,B)} w_j = w_i.\]

For additional information on matroids and the tropical geometry of linear subspaces, we refer to [ Reference Maclagan and SturmfelsMS15 , chapter 4].

4·1. Proof of Proposition 4·6

Let U be a closed subscheme of T, let $u \in M$ , let $V(\chi^u-1)$ be the closed subscheme of T defined by $\chi^u-1 \in k[M]$ , let $w \in u^\perp \cap N$ , and let $v \in N$ be such that $\ell = \langle u, v \rangle > 0$ and such that $\text{in}_w\ U$ is invariant under the $(\mu_\ell, v)$ -action. Proposition 4·6 is clear when $w=0$ , so we assume that $w \neq 0$ .

Let $\mathcal{O}_w = \text{Spec}\!\left(k\!\left[w^\perp \cap M\right]\right)$ , and let $T \to \mathcal{O}_w$ be the algebraic group homomorphism induced by the inclusion $k\!\left[w^\perp \cap M\right] \to k[M]$ .

Proof. By definition,

\[\text{supp}(\text{in}_w\ f) = \{u^{\prime} \in \text{supp}(f) \mid \langle u^{\prime}, w \rangle = \text{trop}(f)(w)\}.\]

If $f=0$ , the statement is obvious. Thus we may assume that there exists $u^{\prime} \in M$ such that $-u^{\prime} \in \text{supp}(\text{in}_w\ f)$ . Then we have that

\[\text{in}_w \left(\chi^{u^{\prime}} f\right) = \chi^{u^{\prime}} \text{in}_w\ f \in k\!\left[w^\perp \cap M\right].\]

Proof. Let $f_1, \dots, f_m \in k[M]$ be such that $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m \in k[M]$ generate the ideal defining $\text{in}_w\ U$ in T. By Lemma 4·8, we can assume that $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m \in k\!\left[w^\perp \cap M\right]$ . Because $w \in u^\perp$ , we have that $\chi^u \in k\!\left[w^\perp \cap M\right]$ .

Thus we may let Y be the closed subscheme of $\mathcal{O}_w$ defined by the ideal generated by $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m \in k\!\left[w^\perp \cap M\right]$ , and we may let Z be the closed subscheme of $\mathcal{O}_w$ defined by the ideal generated by $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m, \chi^u-1 \in k\!\left[w^\perp \cap M\right]$ , and we are done.

Proof. The composition of the co-character $\mathbb{G}_{m,k} \to T$ corresponding to v with the morphism $T \to \mathcal{O}_w$ corresponds to the map of lattices $w^\perp \cap M \hookrightarrow M \xrightarrow{\langle \cdot, v \rangle} \mathbb{Z}$ , and the composition of the co-character $\mathbb{G}_{m,k} \to T$ corresponding to $v-w$ with the morphism $T \to \mathcal{O}_w$ corresponds to the map of lattices $w^\perp \cap M \hookrightarrow M \xrightarrow{\langle \cdot, v-w \rangle} \mathbb{Z}$ . These are clearly the same lattice maps, so we are done.

Let $T_w = \text{Spec}(k[(\mathbb{R} w \cap N)^\vee])$ . Any splitting of $0 \to \mathbb{R} w \cap N \to N \to N/(\mathbb{R} w \cap N) \to 0$ induces an isomorphism of algebraic groups $T \cong T_w \times_k \mathcal{O}_w$ such that $T \to \mathcal{O}_w$ corresponds to the projection $T_w \times_k \mathcal{O}_w \to \mathcal{O}_w$ .

Let $\phi_1\,:\, \mu_\ell \to T$ (resp. $\phi_2\,:\, \mu_\ell \to T$ ) be the composition of $\mu_\ell \hookrightarrow \mathbb{G}_{m,k}$ with the co-character $\mathbb{G}_{m,k} \to T$ corresponding to v (resp. $v-w$ ).

Let $\varphi_1\,:\, \mu_\ell \to \mathcal{O}_w$ (resp. $\varphi_2\,:\, \mu_\ell \to \mathcal{O}_w$ ) be the composition of $\phi_1$ (resp. $\phi_2$ ) with $T \to \mathcal{O}_w$ .

Proof. This follows directly from Lemma 4·10.

Let $\psi_1\,:\, \mu_\ell \to T_w$ (resp. $\psi_2\,:\, \mu_\ell \to T_w$ ) be the composition of $\phi_1$ (resp. $\phi_2$ ) with the projection $T \cong T_w \times_k \mathcal{O}_w \to T_w$ .

We now prove the first part of Proposition 4·6.

Proof. By Proposition 4·9, there exists a closed subscheme Y of $\mathcal{O}_w$ such that $\text{in}_w\ U$ is equal to the pre-image of Y under the morphism $T \to \mathcal{O}_w$ . Then under the identification $T \cong T_w \times_k \mathcal{O}_w$ , we have that

\[\text{in}_w\ U = T_w \times_k Y.\]

Because $\text{in}_w\ U$ is invariant under the $(\mu_\ell, v)$ -action, Remark 4·12 implies that Y is invariant under the $\mu_\ell$ -action on $\mathcal{O}_w$ induced by $\varphi_1$ . By Lemma 4·11, Y is invariant under the $\mu_\ell$ -action on $\mathcal{O}_w$ induced by $\varphi_2$ , and by Remark 4·12, this implies that $\text{in}_w\ U$ is invariant under the $(\mu_\ell, v-w)$ -action.

Before we complete the proof of Proposition 4·6, we make the following observation, which follows from [ Reference Kutler and UsatineKU22 , lemma 7·1] and the fact that $\dim T_w = 1$ .

We now complete the proof of Proposition 4·6.

Proposition 4·15. We have that

\[\left[\left(V\!\left(\chi^u-1\right) \cap \text{in}_w\ U\right)_\ell^v, \mu_\ell\right] = \left[ \left(V\!\left(\chi^u-1\right) \cap \text{in}_w\ U\right)_\ell^{v-w}, \mu_\ell\right] \in K_0^{\mu_\ell}(\textbf{Var}_k).\]

Proof. By Proposition 4·9, there exists a closed subscheme Z of $\mathcal{O}_w$ such that $V(\chi^u-1) \cap \text{in}_w\ U$ is equal to the pre-image of Z under the morphism $T \to \mathcal{O}_w$ . Then under the identification $T \cong T_w \times_k \mathcal{O}_w$ , we have that

\[V(\chi^u-1) \cap \text{in}_w\ U = T_w \times_k Z.\]

Because $V(\chi^u-1) \cap \text{in}_w\ U$ is invariant under the $(\mu_\ell, v)$ -action, Remark 4·12 implies that Z is invariant under the $\mu_\ell$ -action on $\mathcal{O}_w$ induced by $\varphi_1$ .

Now endow Z with the $\mu_\ell$ -action given by restriction of the $\mu_\ell$ -action on $\mathcal{O}_w$ induced by $\varphi_1$ , which by Lemma 4·11 is the same as the $\mu_\ell$ -action given by restriction of the $\mu_\ell$ -action on $\mathcal{O}_w$ induced by $\varphi_2$ . Then by Remarks 4·12 and 4·14,

\begin{align*} \left[\left(V\!\left(\chi^u-1\right) \cap \text{in}_w\ U\right)_\ell^v, \mu_\ell\right] &= (\mathbb{L}-1)[Z, \mu_\ell]\\ &= \left[\left(V\!\left(\chi^u-1\right) \cap \text{in}_w\ U\right)_\ell^{v-w}, \mu_\ell\right].\end{align*}

5·1. Proof of Corollary 5·4

The following proposition will be used for Corollary 5·4(c). Its purpose is to show that $Z^{\hat\mu}_{X,u,0}$ is computed by summing over lattice points in $\text{Trop}(U)$ with strictly positive entries.

Proof. This is a direct consequence of the X-scheme structure of $\text{in}_w\ U$ .

Now, using the notation in the theorem’s statement, Theorem 5·2 implies

\[Z^{\hat\mu}_{X,u,k}(T) = \sum_{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)} \left[(V_u \cap \text{in}_w\ U)^w_{u \cdot w}, \hat\mu\right] \mathbb{L}^{-d-\text{ordjac}(w)} T^{u \cdot w},\]

and if in addition, the origin of $\mathbb{A}_k^n$ is contained in X, Proposition 5·5 and Theorem 5·2 imply

\[Z^{\hat\mu}_{X,u,0}(T) = \sum_{w \in \text{Trop}(U) \cap \mathbb{Z}_{> 0}^n} \left[(V_u \cap \text{in}_w\ U)^w_{u \cdot w}, \hat\mu\right] \mathbb{L}^{-d-\text{ordjac}(w)} T^{u \cdot w}.\]

Thus Corollary 5·4 follows from Theorem 5·2 and the following proposition.

Proposition 5·6. Suppose there exists $v \in \mathbb{Z}^n$ such that $u \cdot v > 0$ and such that for all $w \in \mathbb{Z}^n$ ,

\[\text{in}_w\ U = \text{in}_{w+v}\ U.\]

Let $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ , and let $V_u$ be the subscheme of $\mathbb{G}_{m,k}^n$ defined by $(x_1, \dots, x_n)^u-1$ . Then $V_u \cap \text{in}_w\ U$ is invariant under the $(\mu_{u \cdot v}, v)$ -action and

\[\left[(V_u \cap \text{in}_w\ U)^w_{u \cdot w}, \hat\mu\right] = [(V_u \cap \text{in}_w\ U)^v_{u \cdot v}, \hat\mu] \in K_0^{\hat\mu}(\textbf{Var}_k).\]

Proof. Because $u \cdot w > 0$ , there exist $\ell, \ell^{\prime} \in \mathbb{Z}_{>0}$ and $w^{\prime} \in \mathbb{Z}^n$ such that $u \cdot w^{\prime} = 0$ and

\[\ell w = \ell^{\prime} v + w^{\prime}.\]

By Proposition 4·5, $V_u \cap \text{in}_w\ U$ is invariant under the $(\mu_{u \cdot \ell w}, \ell w)$ -action and

\[\left[(V_u \cap \text{in}_w\ U)^w_{u \cdot w}, \hat\mu\right] = \left[(V_u \cap \text{in}_w\ U)^{\ell w}_{u \cdot \ell w}, \hat\mu\right] \in K_0^{\hat\mu}(\textbf{Var}_k).\]

By the hypotheses on v, we have that

\[\text{in}_{w^{\prime}}\ U = \text{in}_{\ell w}\ U,\]

so by Proposition 4·3, $\text{in}_{w^{\prime}}\ U$ is invariant under the $(\mu_{u \cdot \ell w}, \ell w)$ -action. Then by Proposition 4·6, $\text{in}_{w^{\prime}}\ U$ is invariant under the $(\mu_{u \cdot \ell^{\prime} v}, \ell^{\prime} v)$ -action, and noting that $u \cdot \ell w = u \cdot \ell^{\prime} v$ ,

\[\left[(V_u \cap \text{in}_{w^{\prime}}\ U)^{\ell w}_{u \cdot \ell w}, \mu_{u \cdot \ell w}\right] = \left[(V_u \cap \text{in}_{w^{\prime}}\ U)^{\ell^{\prime} v}_{u \cdot \ell^{\prime} v}, \mu_{u \cdot \ell^{\prime} v}\right] \in K_0^{\mu_{u \cdot \ell^{\prime} v}}(\textbf{Var}_k).\]

Again by Proposition 4·5, $V_u \cap \text{in}_{w^{\prime}}\ U$ is invariant under the $(\mu_{u \cdot v}, v)$ -action and

\[\left[(V_u \cap \text{in}_{w^{\prime}}\ U)^{\ell^{\prime} v}_{u \cdot \ell^{\prime} v}, \hat\mu\right] = \left[(V_u \cap \text{in}_{w^{\prime}}\ U)^v_{u \cdot v}, \hat\mu\right] \in K_0^{\hat\mu}(\textbf{Var}_k).\]

All together, noting that $\text{in}_w\ U = \text{in}_{\ell w}\ U = \text{in}_{w^{\prime}}\ U$ ,

\begin{align*} \left[(V_u \cap \text{in}_w\ U)^w_{u \cdot w}, \hat\mu\right] &= \left[(V_u \cap \text{in}_w\ U)^{\ell w}_{u \cdot \ell w}, \hat\mu\right]\\ &= \left[(V_u \cap \text{in}_{w^{\prime}}\ U)^{\ell w}_{u \cdot \ell w}, \hat\mu\right]\\ &= \left[(V_u \cap \text{in}_{w^{\prime}}\ U)^{\ell^{\prime} v}_{u \cdot \ell^{\prime} v}, \hat\mu\right]\\ &= \left[(V_u \cap \text{in}_{w^{\prime}}\ U)^v_{u \cdot v}, \hat\mu\right]\\ &= \left[(V_u \cap \text{in}_w\ U)^v_{u \cdot v}, \hat\mu\right].\end{align*}

5·2. Fibers of tropicalisation

For the remainder of Section 5, fix $\ell \in \mathbb{Z}_{>0}$ and endow R with the $\mu_\ell$ -action where each $\xi \in \mu_{\ell}$ acts on R by the $\pi$ -adically continuous k-morphism $\pi \mapsto \xi^{-1}\pi$ . Let $\mathbb{A}_R^n = \text{Spec}(R[x_1, \dots, x_n])$ , let $\mathfrak{X} = X \times_k \text{Spec}(R) \subset \mathbb{A}_R^n$ , and endow $\mathbb{A}_R^n$ (resp. $\mathfrak{X}$ ) with the $\mu_\ell$ -action induced by the $\mu_\ell$ -action on R and the trivial $\mu_\ell$ -action on $\mathbb{A}_k^n$ (resp. X).

Let $A_\ell \subset \mathscr{G}(\mathfrak{X})$ be the subset of arcs where $(x_1, \dots, x_n)^u|_\mathfrak{X}$ has order $\ell$ , and let $A_{\ell, 1} \subset \mathscr{G}(\mathfrak{X})$ be the subset of arcs where $(x_1, \dots, x_n)^u|_\mathfrak{X}$ has order $\ell$ and angular component 1.

Let $\text{trop}\,:\, \mathscr{G}(\mathfrak{X}) \to (\mathbb{Z}_{\geq 0} \cup \{\infty\})^n$ be the function $\left(\text{ord}_{x_1|_\mathfrak{X}}, \dots, \text{ord}_{x_n|_\mathfrak{X}}\right)$ . Any arc that tropicalises to a point in $\mathbb{Z}_{\geq 0}^n$ has generic point in $U \times_k \text{Spec}(R)$ , so

\[\text{trop}(\mathscr{G}(\mathfrak{X})) \cap \mathbb{Z}_{\geq 0}^n \subset \text{Trop}(U).\]

Also because $u \in \mathbb{Z}_{>0}^n$ and $\ell \neq 0$ ,

\[\text{trop}(A_\ell) \subset \text{trop}(\mathscr{G}(\mathfrak{X})) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right) \subset \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right).\]

Thus

\[A_\ell = \bigcup_{\substack{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)\\ u \cdot w = \ell}} \text{trop}^{-1}(w).\]

This union is disjoint, and because $u \in \mathbb{Z}_{> 0}^n$ , it is also finite. By Proposition 3·4, for each $w \in \mathbb{Z}_{\geq 0}^n$ , we have that the fiber $\text{trop}^{-1}(w)$ and the intersection $\text{trop}^{-1}(w) \cap A_{\ell, 1}$ are $\mu_\ell$ -invariant cylinders in $\mathscr{G}(\mathfrak{X})$ . We have thus proved the following.

Proposition 5·7. We have that

\[\mu_{\mathfrak{X}}^{\mu_\ell}(A_{\ell, 1}) = \sum_{\substack{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)\\ u \cdot w = \ell}} \mu_{\mathfrak{X}}^{\mu_\ell} \left(\text{trop}^{-1}(w) \cap A_{\ell, 1}\right),\]

and

\[\mu_{\mathfrak{X}}(A_\ell) = \sum_{\substack{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)\\ u \cdot w = \ell}} \mu_{\mathfrak{X}}\left(\text{trop}^{-1}(w)\right).\]

5·3. Morphisms for computing volumes

Throughout subsection 5·3, we will fix some $w = (w_1, \dots, w_n) \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ such that $u \cdot w = \ell$ . We will construct a smooth, pure relative dimension d, finite type, separated R-scheme $\mathfrak{X}^w$ with good $\mu_\ell$ -action making the structure morphism equivariant, and we will construct a $\mu_\ell$ -equivariant morphism $\psi_w\,:\, \mathfrak{X}^w \to \mathfrak{X}$ that will eventually be used to compute the motivic volumes of $\text{trop}^{-1}(w) \cap A_{\ell, 1}$ and $\text{trop}^{-1}(w)$ .

Let $\mathbb{G}_{m,R}^n = \text{Spec}\!\left(R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]\right) = \mathbb{G}_{m,k}^n \times_k \text{Spec}(R)$ , and endow it with the $\mu_\ell$ -action induced by the $\mu_\ell$ -action on $\text{Spec}(R)$ and the $(\mu_\ell, w)$ -action on $\mathbb{G}_{m,k}^n$ . Let $\varphi_w\,:\, \mathbb{G}_{m,R}^n \to \mathbb{A}_R^n$ be the R-scheme morphism corresponding to the R-algebra morphism

\[\varphi_w^*\,:\, \text{Spec}(R[x_1, \dots, x_n]) \to \text{Spec}\!\left(R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]\right)\,:\, x_i \mapsto \pi^{w_i}x_i.\]

Proof. Let $\xi \in \mu_\ell$ , and let $\xi_1\,:\, R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right] \to R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]$ and $\xi_2\,:\, R[x_1, \dots, x_n] \to R[x_1, \dots, x_n]$ be its actions. We will show that

\[\xi_1 \circ \varphi_w^* = \varphi_w^* \circ \xi_2.\]

Because the structure morphisms of $\mathbb{G}_{m,R}^n$ and $\mathbb{A}_R^n$ are $\mu_\ell$ -equivariant, it is sufficient to show that if $i \in \{1, \dots, n\}$ , then

\[\xi_1\left(\varphi_w^*(x_i)\right) = \varphi_w^*(\xi_2(x_i)),\]

which holds because

\begin{align*} \xi_1\left(\varphi_w^*(x_i)\right) &= \xi_1\left(\pi^{w_i} x_i\right)\\ &= \left(\xi^{-1} \pi\right)^{w_i}\xi^{w_i} x_i\\ &= \pi^{w_i} x_i\\ &= \varphi_w^*(x_i)\\ &= \varphi_w^*(\xi_2(x_i)).\end{align*}

Now let $\mathfrak{X}_\eta$ be the generic fiber of $\mathfrak{X}$ , let $\varphi_{w,\eta}\,:\, \mathbb{G}_{m,K}^n \to \mathbb{A}_{K}^n$ be the base change of $\varphi_w$ to the fraction field K of R, let $\mathfrak{X}^w_\eta \subset \mathbb{G}_{m,K}^n$ be the pre-image of $\mathfrak{X}_\eta$ under $\varphi_{w,\eta}$ , and let $\mathfrak{X}^w \subset \mathbb{G}_{m,R}^n$ be the unique closed subscheme of $\mathbb{G}_{m,R}^n$ that is flat over R and has generic fiber $\mathfrak{X}^w_\eta$ , see for example [ Reference GublerGub13 , section 4]. By construction, the generic fiber of $\mathfrak{X}^w$ is isomorphic to $U \times_k \text{Spec}(K)$ , and its special fiber is equal to $\text{in}_w\ U \subset \mathbb{G}_{m,k}^n$ , which is smooth by the hypotheses on X. Thus $\mathfrak{X}^w$ is smooth and pure relative dimension d over R. Note that by uniqueness, $\mathfrak{X}^w$ is equal to the closed subscheme of $\varphi_w^{-1}(\mathfrak{X})$ defined by its R-torsion ideal. Thus we have a morphism $\psi_w\,:\, \mathfrak{X}^w \to \mathfrak{X}$ induced from $\varphi_w$ by restriction.

To obtain a generating set for the ideal defining $\mathfrak{X}^w$ in $\mathbb{G}_{m,R}^n$ , we first need to prove the following lemma.

Proof. We may assume $\mathfrak{Y} = \text{Spec}(\mathfrak{A})$ for some finite type R-algebra $\mathfrak{A}$ . Let $I \subset \mathfrak{A}$ be the $\pi$ -torsion ideal of $\mathfrak{A}$ . Because I is finitely generated, there exists $m \in \mathbb{Z}_{\geq 0}$ such that $\pi^mI = 0$ . By the hypotheses,

\[I \subset \pi \mathfrak{A}.\]

Let $f \in I$ . Then there exists $g \in \mathfrak{A}$ such that $f = \pi g$ . But $\pi g \in I$ implies that $g \in I$ . Thus

\[I = \pi I = \pi^m I = 0.\]

Therefore $\mathfrak{A}$ is $\pi$ -torsion free, so it is flat over R.

We can now prove the following two propositions.

Proposition 5·11. Let $f_1, \dots, f_m \in k[x_1, \dots, x_n]$ be a generating set for the ideal defining X in $\mathbb{A}_k^n$ such that $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m \in k\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]$ form a generating set for the ideal of $\text{in}_w\ U$ in $\mathbb{G}_{m,k}^n$ . Then

\[\pi^{-\text{trop}(f_1)(w)}\varphi_w^*(f_1), \dots, \pi^{-\text{trop}(f_m)(w)}\varphi_w^*(f_m) \in R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]\]

form a generating set for the ideal defining $\mathfrak{X}^w$ in $\mathbb{G}_{m,R}^n$ .

Proof. Let $\mathfrak{Y}$ be the closed subscheme of $\mathbb{G}_{m,R}^n$ defined by the ideal generated by

\[\pi^{-\text{trop}(f_1)(w)}\varphi_w^*(f_1), \dots, \pi^{-\text{trop}(f_m)(w)}\varphi_w^*(f_m).\]

Then by construction, the generic fiber of $\mathfrak{Y}$ is equal to $\mathfrak{X}^w_\eta$ , and $\mathfrak{X}^w$ is equal to the closed subscheme of $\mathfrak{Y}$ defined by its R-torsion ideal. The special fiber of $\mathfrak{Y}$ is the closed subscheme of $\mathbb{G}_{m,k}^n$ defined by $\text{in}_w\ f_1, \dots, \text{in}_w\ f_m$ and thus is equal to $\text{in}_w\ U$ , which is also the special fiber of $\mathfrak{X}^w$ . Therefore by Lemma 5·10, $\mathfrak{Y}$ is flat over R, so $\mathfrak{X}^w$ is equal to $\mathfrak{Y}$ .

Proof. By the hypotheses on X, we know there exist $f_1, \dots, f_{n-d} \in k[x_1, \dots, x_n]$ that generate the ideal of X such that $\text{in}_w\ f_1, \dots, \text{in}_w\ f_{n-d}$ generate the ideal of $\text{in}_w\ U$ , so by Proposition 5·11,

\[\pi^{-\text{trop}(f_1)(w)}\varphi_w^*(f_1), \dots, \pi^{-\text{trop}(f_{n-d})(w)}\varphi_w^*(f_{n-d}) \in R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]\]

generate the ideal defining $\mathfrak{X}^w$ in $\mathbb{G}_{m,R}^n$ .

Thus it will be sufficient to show that if $f \in k[x_1, \dots, x_n]$ , $\xi \in \mu_\ell$ , and $\xi_1\,:\, R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right] \to R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]$ is its action, then $\xi_1(\pi^{-\text{trop}(f)(w)}\varphi_w^*(f))$ is in the ideal of $R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]$ generated by $\pi^{-\text{trop}(f)(w)}\varphi_w^*(f)$ . Write

\[f = \sum_{u^{\prime} \in \mathbb{Z}_{\geq 0}^n} a_{u^{\prime}} (x_1, \dots, x_n)^{u^{\prime}},\]

where each $a_{u^{\prime}} \in k$ . Then

\[\pi^{-\text{trop}(f)(w)}\varphi_w^*(f) = \sum_{u^{\prime} \in \mathbb{Z}_{\geq 0}^n} \pi^{u^{\prime} \cdot w - \text{trop}(f)(w)} a_{u^{\prime}} (x_1, \dots, x_n)^{u^{\prime}},\]

so

\begin{align*} \xi_1(\pi^{-\text{trop}(f)(w)}\varphi_w^*(f)) &= \sum_{{u^{\prime}} \in \mathbb{Z}_{\geq 0}^n} \left(\xi^{-1} \pi\right)^{u^{\prime} \cdot w - \text{trop}(f)(w)} a_{u^{\prime}} \xi^{u^{\prime} \cdot w} (x_1, \dots, x_n)^{u^{\prime}}\\ &= \xi^{\text{trop}(f)(w)}\sum_{u^{\prime} \in \mathbb{Z}_{\geq 0}^n} \pi^{u^{\prime} \cdot w - \text{trop}(f)(w)} a_{u^{\prime}} (x_1, \dots, x_n)^{u^{\prime}}\\ &= \xi^{\text{trop}(f)(w)} \pi^{-\text{trop}(f)(w)} \varphi_w^*(f).\end{align*}

We now endow $\mathfrak{X}^w$ with the restriction of the $\mu_\ell$ -action on $\mathbb{G}_{m,R}^n$ . Because $\mathfrak{X}^w$ is affine, this $\mu_\ell$ -action is good, and by construction, this $\mu_\ell$ -action makes the structure morphism equivariant. By Proposition 5·8, we have that the morphism $\psi_w\,:\, \mathfrak{X}^w \to \mathfrak{X}$ is $\mu_\ell$ -equivariant.

5·4. Preparing for the change of variables formula

We establish some results which will be used in the application of Theorem 2·8.

For the remainder of Section 5, let $V_u$ be the subscheme of $\mathbb{G}_{m,k}^n$ defined by $(x_1, \dots, x_n)^u -1$ , and if $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ is such that $u \cdot w = \ell$ , let $\mathfrak{X}^w$ and $\psi_w\,:\, \mathfrak{X}^w \to \mathfrak{X}$ be as constructed in subsection 5·3.

Proposition 5·14. Let $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ be such that $u \cdot w = \ell$ . Noting that $V_u \cap \text{in}_w\ U \subset \text{in}_w\ U = \mathfrak{X}^w_0$ , the subset $\theta_0^{-1}(V_u \cap \text{in}_w\ U) \subset \mathscr{G}(\mathfrak{X}^w)$ is a $\mu_\ell$ -invariant cylinder, and

\[\mu_{\mathfrak{X}^w}^{\mu_\ell}\left(\theta_0^{-1}\left(V_u \cap \text{in}_w\ U\right)\right) = \left[\left(V_u \cap \text{in}_w\ U\right)^w_\ell / \mathfrak{X}^w_0, \mu_\ell\right] \mathbb{L}^{-d} \in \mathscr{M}^{\mu_\ell}_{\mathfrak{X}^w_0}.\]

Proof. By Proposition 4·4 and Remark 5·13, we have that $V_u \cap \text{in}_w\ U$ is a $\mu_\ell$ -invariant subscheme of $\mathfrak{X}^w_0$ , and with the restriction of this $\mu_\ell$ -action, it is equal to $(V_u \cap \text{in}_w\ U)^w_\ell$ . The proposition then follows from the fact that the truncation morphism $\theta_0\,:\, \mathscr{G}(\mathfrak{X}^w) \to \mathscr{G}_0(\mathfrak{X}^w) = \mathfrak{X}^w_0$ is $\mu_\ell$ -equivariant and the definition of the $\mu_\ell$ -equivariant motivic measure.

Lemma 5·15. Let $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ be such that $u \cdot w = \ell$ , let $\varphi_w\,:\, \mathbb{G}_{m,R}^n \to \mathbb{A}_R^n$ be as in subsection 5·3, and let k^′ be an extension of k. Then $\varphi_w$ induces a bijection

\[\mathscr{G}\!\left( \mathbb{G}_{m,R}^n \right)(k^{\prime}) \to \left\{ x \in \mathscr{G}\!\left(\mathbb{A}_R^n\right)(k^{\prime}) \mid w = \left(\text{ord}_{x_1}(x), \dots, \text{ord}_{x_n}(x)\right)\right\}.\]

Proof. Let $R^{\prime} = k^{\prime}[\![ \pi ]\!]$ , and let K ^′ be its field of fractions. Because $\varphi_w$ induces an open immersion on generic fibers, it induces an injection $\mathbb{G}_{m,R}^n(K^{\prime}) \to \mathbb{A}_R^n(K^{\prime})$ . Because $\mathbb{G}^n_{m,R}$ is separated, this implies that $\varphi_w$ induces an injection $\mathscr{G}\!\left(\mathbb{G}_{m,R}^n\right)(k^{\prime}) \to \mathscr{G}\!\left(\mathbb{A}_R^n\right)(k^{\prime})$ . We thus only need to show that the image of this injection is $\left\{ x \in \mathscr{G}\!\left(\mathbb{A}_R^n\right)(k^{\prime}) \mid w = (\text{ord}_{x_1}(x), \dots, \right.$ $\left. \text{ord}_{x_n}(x))\right\}$ .

Let $y\,:\, \text{Spec}(R^{\prime}) \to \mathbb{G}_{m,R}^n$ . Then for each $i \in \{1, \dots, n\}$ , we have that $x_i(y)$ is a unit in R ^′, so by construction,

\[\varphi_w(y) \in \left\{ x \in \mathscr{G}\!\left(\mathbb{A}_R^n\right)(k^{\prime}) \mid w = \left(\text{ord}_{x_1}(x), \dots, \text{ord}_{x_n}(x)\right)\right\}.\]

Write $w = (w_1, \dots, w_n)$ , and let $x \,:\, \text{Spec}(R^{\prime}) \to \mathbb{A}_R^n$ be such that $\text{ord}_{x_i}(x) = w_i$ for each $i \in \{1, \dots, n\}$ . Then for each $i \in \{1, \dots, n\}$ , we have that $\pi^{-w_i} x_i(x)$ is a unit in R ^′, so we may set $y\,:\, \text{Spec}(R^{\prime}) \to \mathbb{G}_{m,R}^n$ to be the morphism whose pullback is given by $x_i \mapsto \pi^{-w_i}x_i(x) \in R$ . By construction $\varphi_w(y) = x$ , and we are done.

Proof. Fix an extension k ^′ of k. Because $\psi_w$ induces an open immersion on generic fibers and because $\mathfrak{X}^w$ is separated, we have that $\varphi_w$ induces an injection from $\mathscr{G}(\mathfrak{X}^w)(k^{\prime})$ to $\mathscr{G}(\mathfrak{X})(k^{\prime})$ . Thus we need to show that the image of $\mathscr{G}(\mathfrak{X}^w)(k^{\prime})$ is $\text{trop}^{-1}(w)(k^{\prime})$ and that the image of $\theta_0^{-1}(V_u \cap \text{in}_w\ U)(k^{\prime})$ is $(\text{trop}^{-1}(w) \cap A_{\ell, 1})(k^{\prime})$ .

Let $y \in \mathscr{G}(\mathfrak{X}^w)(k^{\prime}) \subset \mathscr{G}\!\left(\mathbb{G}_{m,R}^n\right)(k^{\prime})$ . By Lemma 5·15, $\psi_w(y) \in \text{trop}^{-1}(w)(k^{\prime})$ . Let $x \in \text{trop}^{-1}(w)(k^{\prime}) \subset \left\{ x^{\prime} \in \mathscr{G}\!\left(\mathbb{A}_R^n\right)(k^{\prime}) \mid w = \left(\text{ord}_{x_1}(x^{\prime}), \dots, \text{ord}_{x_n}(x^{\prime})\right)\right\}$ . By Lemma 5·15, x is in the image of $\varphi_w$ , where $\varphi_w$ is as in Subsection 5·3. Because $\mathfrak{X}^w$ is the closed subscheme of $\varphi_w^{-1}(\mathfrak{X})$ defined by its R-torsion ideal, this implies that x is in the image $\psi_w$ . Thus $\psi_w$ induces a bijection $\mathscr{G}(\mathfrak{X}^w)(k^{\prime}) \to \text{trop}^{-1}(w)(k^{\prime})$ .

Let $y \in \mathscr{G}(\mathfrak{X}^w)(k^{\prime})$ . We only need to show that $\psi_w(y) \in A_{\ell, 1}(k^{\prime})$ if and only if $\theta_0(y) \in (V_u \cap \text{in}_w\ U)(k^{\prime})$ . Write $w = (w_1, \dots, w_n)$ , and let $R^{\prime} = k^{\prime} [\![ \pi ]\!]$ . Then

\begin{align*} \psi_w(y) \in A_{\ell, 1}(k^{\prime}) &\iff (x_1, \dots, x_n)^u(\psi_w(y)) = \pi^{\ell}(1+\pi r) \text{ for some $r \in R^{\prime}$}\\ &\iff (\pi^{w_1} x_1, \dots, \pi^{w_n} x_n)^u (y) = \pi^{u \cdot w}(1 + \pi r) \text{ for some $r \in R^{\prime}$}\\ &\iff (x_1, \dots, x_n)^u(y) = 1 + \pi r \text{ for some $r \in R^{\prime}$}\\ &\iff ((x_1, \dots, x_n)^u - 1)(\theta_0(y)) = 0\\ &\iff \theta_0(y) \in (V_u \cap \text{in}_w\ U)(k^{\prime}).\end{align*}

Proposition 5·17. Let $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ be such that $u \cdot w = \ell$ , and let $f_1, \dots, f_{n-d} \in k[x_1, \dots, x_n]$ be a generating set for the ideal defining X in $\mathbb{A}_k^n$ such that $\text{in}_w\ f_1, \dots, \text{in}_w\ f_{n-d} \in k\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]$ form a generating set for the ideal of $\text{in}_w\ U$ in $\mathbb{G}_{m,k}^n$ . Then the jacobian ideal of $\psi_w$ is generated by

\[\pi^{w_1+\dots+w_n-(\text{trop}(f_1)(w)+\dots+\text{trop}(f_{n-d})(w))}.\]

Proof. Let $\varphi_w\,:\, \mathbb{G}_{m,R}^n \to \mathbb{A}_R^n$ be as in Subsection 5·3, and for any $f \in R[x_1, \dots, x_n]$ , we will set

\[f^w = \pi^{-\text{trop}(f)(w)}\varphi_w^*(f) \in R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right].\]

Then by Proposition 5·11, the ideal defining $\mathfrak{X}^w$ is generated by $f_1^w, \dots, f_{n-d}^w$ . Let $\mathfrak{A}_w= R\!\left[x_1^{\pm 1}, \dots, x_n^{\pm 1}\right]/\left(f_1^w, \dots, f_{n-d}^w\right)$ be the coordinate ring of $\mathfrak{X}^w$ . Then we have the diagram

where the right vertical sequence is the presentation for the differentials module $\Omega_{\mathfrak{X}^w / R}$ induced by our presentation for $\mathfrak{A}_w$ , the top horizontal sequence is the standard presentation for the relative differentials module $\Omega_{\mathfrak{X}^w / \mathfrak{X}}$ , and the top left vertical arrow picks out the generators of $\psi_w^*\Omega_{\mathfrak{X}, R}$ induced by the coordinates of $\mathbb{A}_R^n$ . The diagonal sequence gives a presentation for $\Omega_{\mathfrak{X}^w / \mathfrak{X}}$ with matrix whose entries are the images in $\mathfrak{A}_w$ of the entries in the matrix

\[\begin{pmatrix}\pi^{w_1} & \quad & \quad & \quad \partial f_1^w/\partial x_1 & \quad & \quad \partial f_{n-d}^w/ \partial x_1\\[3pt]& \quad \ddots & \quad & \quad \vdots & \quad \cdots & \quad \vdots\\[3pt]& \quad & \quad \pi^{w_n} & \quad \partial f_1^w/\partial x_n & \quad & \quad \partial f_{n-d}^w/\partial x_n\end{pmatrix}.\]

For each $i \in \{1,\dots, n\}$ and $j \in \{1, \dots, n-d\}$ ,

\[\partial \varphi_w^*(f_j)/ \partial x_i = \sum_{i^{\prime}=1}^n \left(\varphi_w^*\left(\partial f_j / \partial x_{i^{\prime}}\right)\right) \left(\partial \varphi_w^*\left(x_{i^{\prime}}\right) /\partial x_i\right) = \pi^{w_i} \varphi_w^*\left(\partial f_j / \partial x_i\right),\]

so

\[\partial f_j^w/\partial x_i = \pi^{w_i - \text{trop}(f_j)(w)} \varphi_w^*\left( \partial f_j / \partial x_i\right).\]

For each $m \in \{0, 1, \dots, n-d\}$ and size m subsets $I \subset \{1, \dots, n\}$ and $J \subset \{1, \dots, n-d\}$ , let $\Delta_I^J$ be the determinant of the size m minor of the matrix $(\partial f_j / \partial x_i)_{i,j}$ given by rows in I and columns in J. Then the jacobian ideal of $\psi_w$ is generated by the images in $\mathfrak{A}_w$ of

\[\left\{\pi^{w_1 + \dots + w_n - \sum_{j \in J}\text{trop}(f_j)(w)} \varphi_w^* \Delta_I^J\right\}_{m, I, J}.\]

Because $\mathfrak{X}$ is smooth and pure relative dimension d over R, the unit ideal of $\mathfrak{X}$ is generated by the images in $R[x_1, \dots, x_n]/(f_1, \dots, f_{n-d})$ of

\[\left\{\Delta_I^{\{1, \dots, n-d\}} \mid \text{$I \subset \{1, \dots, n\}$ has size $n-d$}\right\},\]

so the jacobian ideal of $\psi_w$ contains

\[\pi^{w_1+\dots+w_n-(\text{trop}(f_1)(w)+\dots+\text{trop}(f_{n-d})(w))}.\]

Because $w \in \mathbb{Z}_{\geq 0}^n$ and each $f_j \in k[x_1, \dots, x_n]$ , we have that each $\text{trop}(f_j)(w) \geq 0$ . Therefore the jacobian ideal of $\psi_w$ is in fact generated by

\[\pi^{w_1+\dots+w_n-(\text{trop}(f_1)(w)+\dots+\text{trop}(f_{n-d})(w))}.\]

5·5. Proofs of Proposition 5·1 and Theorem 5·2

We first prove Proposition 5·1.

Proof of Proposition 5·1. This is clear if $w = 0$ , so we may assume that $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ is such that $u \cdot w = \ell$ . In this case, by Remark 5·13, the special fiber of $\mathfrak{X}^w$ with its induced $\mu_\ell$ -action is equal to $(\text{in}_w\ U)^w_\ell$ . The proposition follows by considering the special fiber of $\psi_w\,:\, \mathfrak{X}^w \to \mathfrak{X}$ , where by construction $\psi_w$ is the restriction of $\varphi_w$ and the special fiber of $\varphi_w$ is precisely the map $\varphi$ from the statement of Proposition 5·1.

Now set $\text{ordjac}\,:\, \text{Trop}(U) \cap (\mathbb{Z}^n_{\geq 0} \setminus \{0\}) \to \mathbb{Z}\,:\, w \mapsto \text{ordjac}_{\psi_w}(y)$ for any $y \in \mathscr{G}(\mathfrak{X}^w)$ , noting that by Proposition 5·17, this does not depend on the choice of y. Also set $\text{ordjac}(0) = 0$ .

Proposition 5·18. Let $w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)$ be such that $u \cdot w = \ell$ . Then

\[\mu_{\mathfrak{X}}\left(\text{trop}^{-1}(w)\right) = [\text{in}_w\ U / X] \mathbb{L}^{-d - \text{ordjac}(w)},\]

and

\[\mu_{\mathfrak{X}}^{\mu_\ell}\left(\text{trop}^{-1}(w) \cap A_{\ell, 1}\right) = \left[\left(V_u \cap \text{in}_w\ U\right)^w_{u \cdot w}/X, \mu_\ell\right]\mathbb{L}^{-d-\text{ordjac}(w)}.\]

Proof. By Remark 5·9 and Propositions 5·14 and 5·16, the proposition follows from the (equivariant) motivic change of variables formula applied to $\psi_w$ .

Because $u \in \mathbb{Z}_{>0}^n$ , the monomial $(x_1, \dots, x_n)^u$ vanishes on all of $X \setminus U$ . Thus the constant term of $Z_{X,u}^{\textrm{naive}}(T)$ is equal to

\[[U / X]\mathbb{L}^{-d} = [\text{in}_0\ U / X] \mathbb{L}^{-d - \text{ordjac}(w)}.\]

Therefore, Theorem 5·2 follows from Proposition 5·17 and the next proposition.

Proposition 5·19. The coefficient of $T^\ell$ in $Z_{X,u}^{\hat\mu}(T)$ is equal to

\[\sum_{\substack{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)\\ u \cdot w = \ell}} [(V_u \cap \text{in}_w\ U)^w_{u \cdot w} / X, \hat\mu] \mathbb{L}^{-d - \text{ordjac}(w)},\]

and the coefficient of $T^\ell$ in $Z_{X,u}^{\textrm{naive}}(T)$ is equal to

\[\sum_{\substack{w \in \text{Trop}(U) \cap \left(\mathbb{Z}_{\geq 0}^n \setminus \{0\}\right)\\ u \cdot w = \ell}} [\text{in}_w\ U / X] \mathbb{L}^{-d - \text{ordjac}(w)}.\]

Proof. This follows from Propositions 3·5, 3·6, 5·7 and 5·18.