# Grothendieck rings of definable subassignments
and equivariant motivic measures

Lê Quy Thuong  Dedicated to Professors Hà Huy Vui and Ta Lê Loi on the special
occasion of their birthdays University of Science, Vietnam National
University, Hanoi
334 Nguyen Trai street, Thanh Xuan district, Hanoi, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

The paper studies categories of definable subassignments with some category
equivalences to semi-algebraic and constructible subsets of arc spaces of
algebraic varieties. These materials allow us to compare the motivic measure
of Cluckers-Loeser and the one of Denef-Loeser in certain classes of definable
subassignments.

###### Key words and phrases:

motivic measure, definable subassignment, measurable subassignment

###### 2010 Mathematics Subject Classification:

Primary 03C10, 14E18, 14G10

The first author’s research is funded by the Vietnam National University,
Hanoi (VNU) under project number QG.19.06.

##  1\. Introduction

Since it was invented by Kontsevich at the 1995 Orsay seminar, geometric
motivic integration has attained a full development and has become one of the
central objects of algebraic geometry. From algebraic varieties to formal
schemes, the development records contributions of several authors, such as
Denef-Loeser [7, 8], Sebag [18], Loeser-Sebag [14], Nicaise-Sebag [16, 15],
Nicaise [15]. Another point of view on motivic integration known as arithmetic
motivic integration was developed almost at the same time, which works over
only $p$-adic fields (see [9]). Cluckers-Loeser’s motivic integration [3, 4,
5], which was built on model theory with respect to the Denef-Pas languages,
is a general theory of motivic integration. It allows to specialize to both of
arithmetic and geometric points of view (see [10], [6], [1]). This theory of
motivic integration has an important application to the fundamental lemma (see
[2]).

Recently, also using model theory with different languages, Hrushovski-Kazhdan
[11] and Hrushovski-Loeser [12] have extended geometric motivic integration to
the arithmetic aspect, with many interesting results and applications.

Throughout the paper, the ground field $k$ will be always a field of
characteristic zero. The present paper discusses the motivically measurable
subassignments in the formalism of Cluckers-Loeser for motivic integration [3,
4, 5], it also provides a comparison of their measure with the classical
motivic measure of Denef-Loeser [7, 8]. By this purpose, we concentrate on a
special Denef-Pas language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}$ with the
Presburger language for value group sort, and consider the theory
$T_{\mathrm{acl}}$ of algebraically closed fields containing $k$. Let
$\mathrm{Field}_{k}$ be the category of all fields $K$ containing $k$, and
$\mathrm{Field}_{k}(T_{\mathrm{acl}})$ the category of fields $K$ over $k$
such that each $(K(\\!(t)\\!),K,\mathbb{Z})$ is a model of $T_{\mathrm{acl}}$.
For $K$ in $\mathrm{Field}_{k}$, we consider the natural valuation map
$\mathrm{ord}_{t}:K(\\!(t)\\!)^{\times}\to\mathbb{Z}$ augmented by
$\mathrm{ord}_{t}(0)=+\infty$, and the natural angular component map
$\overline{\mathrm{ac}}:K(\\!(t)\\!)^{\times}\to K$ augmented by
$\overline{\mathrm{ac}}(0)=0$. A basic affine subassignment $h[m,n,r]$ (or, in
another notation,
$h_{\mathbb{A}_{k(\\!(t)\\!)}^{m}\times\mathbb{A}_{k}^{n}\times\mathbb{Z}^{r}}$)
is a functor $K\mapsto K(\\!(t)\\!)^{m}\times K^{n}\times\mathbb{Z}^{r}$ from
$\mathrm{Field}_{k}$ to the category of sets. A definable subsassignment of
$h[m,n,r]$ is a set of points in $h[m,n,r]$ satisfying a given formula
$\varphi$; it is not a functor in general. In the first half of this paper, we
study the categories concerning definable $T_{\mathrm{acl}}$-subsassignments
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
in which the language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$ is an
extension of $\mathcal{L}_{\mathrm{DP},\mathrm{P}}$ by adding constants in $k$
so that all polynomials in both valued field sort and residue field sort
having coefficients in $k$.

The category
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
has objects to be small definable $T_{\mathrm{acl}}$-subassignments, which is
comparable with the category $\mathrm{SA}_{k}$ of semi-algebraic subsets of
the arc space of an algebraic $k$-variety. When fixing a $k$-variety $X$ we
get the subcategory
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
of $\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k))$ whose objects
are all the small definable $T_{\mathrm{acl}}$-subassignments of
$h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)}$. The definition of
$\mathrm{SA}_{k}$ and $\mathrm{SA}_{k}(X)$ is given in Section 3.1. The first
main result of this paper is as follows.

######  Theorem 1 (Theorem 4.1).

The categories
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{SA}_{k}$ are equivalent. If $X$ is an algebraic $k$-variety, then
the categories
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{SA}_{k}(X)$ are equivalent.

Let $S$ be an affine $k$-variety, and let
$\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
be the category whose objects $\mathsf{X}\to h_{S}$ are the $h_{S}$-projection
of definable $T_{\mathrm{acl}}$-subassignment $\mathsf{X}$ of $h_{S}\times
h_{\mathbb{A}_{k}^{n}}=h_{S\times_{k}\mathbb{A}_{k}^{n}}$ for some $n$ in
$\mathbb{N}$. In Section 2 we describe the category $\mathrm{Cons}_{S}$ of
constructible morphisms from constructible sets over $k$ to $S$ in which a
morphism in $\mathrm{Cons}_{S}$ from $X\to S$ to $Y\to S$ is determined
uniquely up to $S$-isomorphism on $X$ by the graph of an $S$-morphism $X\to
Y$. This category $\mathrm{Cons}_{S}$ has in fact the same Grothendieck ring
with the category $\mathrm{Var}_{S}$. The second main result of the paper is
stated as follows.

######  Theorem 2 (Theorem 4.2).

For any $k$-variety $S$, the categories
$\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{Cons}_{S}$ are equivalent.

This theorem has several interesting corollaries, such as the following
isomorphism between Grothendieck rings
$K_{0}(\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\cong
K_{0}(\mathrm{Var}_{S})$. Thanks to this isomorphism we can identify the class
$\mathbb{L}$ of the trivial line bundle $S\times_{k}\mathbb{A}_{k}^{1}\to S$
with the class $\big{[}h_{S\times_{k}\mathbb{A}_{k}^{1}}\to h_{S}\big{]}$.
Moreover, if we put

$\mathbb{A}:=\mathbb{Z}\left[\mathbb{L},\mathbb{L}^{-1},\frac{1}{1-\mathbb{L}^{-n}}\mid
n\in\mathbb{N}^{*}\right],$

we have
$K_{0}(\mathrm{RDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\otimes_{\mathbb{Z}[\mathbb{L}]}\mathbb{A}\cong\mathscr{M}_{\mathrm{loc}}$.
We also obtain the monodromic version
$K_{0}^{\hat{\mu}}(\mathrm{RDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\cong
K_{0}^{\hat{\mu}}(\mathrm{Var}_{k})$ and
$K_{0}^{\hat{\mu}}(\mathrm{RDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\otimes_{\mathbb{Z}[\mathbb{L}]}\mathbb{A}\cong\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}$.
Here, $\mathscr{M}_{\mathrm{loc}}$ (resp. $\mathscr{M}_{\mathrm{loc}}^{G}$,
with $G$ is either the group scheme $\mu_{e}$ or
$\hat{\mu}=\varprojlim_{e}\mu_{e}$) is the localization of
$K_{0}(\mathrm{Var}_{k})$ (resp. $K_{0}^{G}(\mathrm{Var}_{k})$) by inverting
$\mathbb{L}$ and $\mathbb{L}^{n}-1$ for all $n$ in $\mathbb{N}^{*}$.

Theorem 10.1.1 of [4] implies that there is a unique functor from the category
of definable subassignments to the category of abelian groups,
$\mathsf{X}\mapsto\mathrm{I}C(\mathsf{X})$, which assigns to $\mathsf{X}\to
h_{\mathrm{Spec}k}$ a group morphism

$\mu:\mathrm{I}C(\mathsf{X})\to\mathscr{M}_{\mathrm{loc}}$

satisfying the axioms A0-A8 in that theorem. By [4, Proposition 12.2.2], if a
definable subassignment $\mathsf{X}$ of $h[m,n,0]$ is bounded (see Section
5.2), the characteristic function $\mathbf{1}_{\mathsf{X}}$ will be in
$\mathrm{I}C(\mathsf{X})$. In this case,
$\mu(\mathsf{X}):=\mu(\mathbf{1}_{\mathsf{X}})$ in
$\mathscr{M}_{\mathrm{loc}}$ is the motivic measure of $\mathsf{X}$. When
$\mathsf{X}$ is an invariant positively bounded definable subassignment of
$h[m,n,0]$ we obtain the following comparison theorem (which also contains
main results of the present paper).

######  Theorem 3 (Theorem 5.4, Proposition 5.6).

Let $\mathsf{X}$ be an invariant definable subassignment of $h[m,n,0]$ such
that, for every $(x,y)$ on $\mathsf{X}$ with $x=(x_{1},\dots,x_{m})$,
$\mathrm{ord}_{t}x_{i}\geq 0$ for all $1\leq i\leq m$. With the notions the
morphism $\mathrm{loc}$ defined in Section 2, $\mathrm{vol}$ in Lemma 5.3 and
$\mathsf{X}[e]$ in the paragraph before Proposition 5.5, for
$e\in\mathbb{N}^{*}$, the following identities hold:

$\displaystyle\mu(\mathsf{X})$
$\displaystyle=\mathrm{loc}(\mathrm{vol}(\mathsf{X}))\qquad\text{in}\ \
\mathscr{M}_{\mathrm{loc}},$ $\displaystyle\mu(\mathsf{X}[e])$
$\displaystyle=\mathrm{loc}(\mathrm{vol}(\mathsf{X}[e]))\quad\text{in}\ \
\mathscr{M}_{\mathrm{loc}}^{\mu_{e}}.$

In fact, for $\mathsf{X}$ small in $h[m,0,0]$, the paper [4] showed early
$\delta(\mu(\mathsf{X}))=\mu^{\prime}(X)$ in $\widehat{\mathscr{M}}_{k}$,
where $\widehat{\mathscr{M}}_{k}$ is a completion of $\mathscr{M}_{k}$ defined
in [8], $\delta$ is the canonical morphism
$\mathscr{M}_{\mathrm{loc}}\to\widehat{\mathscr{M}}_{k}$, $\mu^{\prime}$ is
the Denef-Loeser motivic volume defined in [8], and $X$ is the semi-algebraic
subset of $\mathscr{L}(\mathbb{A}_{k}^{m})$ corresponding to $\mathsf{X}$ via
the equivalence of categories between $\mathrm{SA}_{k}(\mathbb{A}_{k}^{m})$
and
$\mathrm{SDef}_{k}(\mathbb{A}_{k}^{m},\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
in Theorem 4.1.

In the end of this paper, we give a proof of the rationality of the series
$\sum_{e\in\mathbb{N}^{*}}\mu(\mathsf{X}[e])T^{e}$ in
$\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}[[T]]$ with $\mathsf{X}$ an invariant
positively definable subassignment of $h[m,n,0]$.

##  2\. Grothendieck rings of varieties

Let $k$ be a field of characteristic zero, and $S$ an algebraic $k$-variety.
As usual (cf. [7, 8]), we denote by $\mathrm{Var}_{S}$ the category of
$S$-varieties and $K_{0}(\mathrm{Var}_{S})$ its Grothendieck ring. By
definition, $K_{0}(\mathrm{Var}_{S})$ is the quotient of the free abelian
group generated by the $S$-isomorphism classes $[X\to S]$ in
$\mathrm{Var}_{S}$ modulo the following relation

$[X\to S]=[Y\to S]+[X\setminus Y\to S]$

for $Y$ being Zariski closed in $X$. Together with fiber product over $S$,
$K_{0}(\mathrm{Var}_{X})$ is a commutative ring with unity
$1=[\mathrm{Id}:S\to S]$. Put

$\mathbb{L}=[\mathbb{A}_{k}^{1}\times_{k}S\to S]$

and write $\mathscr{M}_{S}$ for the localization of $K_{0}(\mathrm{Var}_{S})$
inverting $\mathbb{L}$. Denote by $\mathscr{M}_{S,\mathrm{loc}}$ the
localization of $\mathscr{M}_{S}$ inverting $\mathbb{L}^{n}-1$ for all $n$ in
$\mathbb{N}^{*}$.

Let $C\mathrm{Var}_{S}$ be the category whose objects are constructible
morphisms from constructible sets over $k$ to $S$ with the set of morphisms
between objects $X\to S$ and $Y\to S$ given by

$\mathrm{Mor}_{C\mathrm{Var}_{S}}(X\to S,Y\to
S):=K_{0}(\mathrm{Var}_{X\times_{S}Y}).$

In other words, a morphism from $X\to S$ to $Y\to S$ in $C\mathrm{Var}_{S}$ is
a finite sum of elements of the form $[U\to X\times_{S}Y]$ (in
$K_{0}(\mathrm{Var}_{X\times_{S}Y})$), with $U$ being an algebraic
$k$-variety. The composition of two basic morphisms $[U\to X\times_{S}Y]$ and
$[V\to Y\times_{S}Z]$ is the following morphism

$[V\to Y\times_{S}Z]\circ[U\to X\times_{S}Y]:=[U\times_{Y}V\to X\times_{S}Z].$

This definition makes sense since the morphism $U\times_{Y}V\to X\times_{S}Z$
commutes with the structural morphisms to $S$, and it can be also extended by
additivity. Clearly, the identity morphism of $X$ in $C\mathrm{Var}_{k}$ is
the class in $K_{0}(\mathrm{Var}_{X\times_{k}X}$ of the diagonal morphism
$X\to X\times_{k}X$.

Denote by $\mathrm{Cons}_{S}$ the subcategory of $C\mathrm{Var}_{S}$ in which
objects of $\mathrm{Cons}_{S}$ are objects of $C\mathrm{Var}_{S}$ and a
morphism of $\mathrm{Cons}_{S}$ from $X\to S$ to $Y\to S$ is an element

$[(\mathrm{Id}_{X},f):X\to X\times_{S}Y]$

in $K_{0}(\mathrm{Var}_{X\times_{S}Y})$. By definition, each morphism of
$\mathrm{Cons}_{S}$ from $X\to S$ to $Y\to S$ is determined uniquely, up to
$S$-automorphism on $X$, by the constructible $S$-morphism of constructible
sets $f:X\to Y$, or alternatively, by the graph of such an $f$. Using the
above definition of Grothendieck ring for the category $\mathrm{Cons}_{S}$ we
get

$K_{0}(\mathrm{Var}_{S})\cong K_{0}(\mathrm{Cons}_{S}).$

Let $X$ be an algebraic $k$-variety, and let $G$ be an algebraic group which
acts on $X$. The $G$-action is called good if every $G$-orbit is contained in
an affine open subset of $X$. Now we fix a good action of $G$ on the
$k$-variety $S$. By definition, the $G$-equivariant Grothendieck group
$K_{0}^{G}(\mathrm{Var}_{S})$ of $G$-equivariant morphisms of $k$-varieties
$X\to S$, where $X$ is endowed with a good $G$-action, is the quotient of the
free abelian group generated by the $G$-equivariant isomorphism classes $[X\to
S,\sigma]$ modulo the following relations

$[X\to S,\sigma]=[Y\to S,\sigma|_{Y}]+[X\setminus Y\to S,\sigma|_{X\setminus
Y}]$

for $Y$ being $\sigma$-stable Zariski closed in $X$, and

$[X\times_{k}\mathbb{A}_{k}^{n}\to S,\sigma]=[X\times_{k}\mathbb{A}_{k}^{n}\to
S,\sigma^{\prime}]$

if $\sigma$ and $\sigma^{\prime}$ lift the same $G$-action on $X$ to an affine
action on $X\times\mathbb{A}_{k}^{n}$. As above, we have the commutative ring
with unity structure on $K_{0}^{G}(\mathrm{Var}_{S})$ by fiber product, where
$G$-action on the fiber product is through the diagonal $G$-action, and we may
define the localization $\mathscr{M}_{S}^{G}$ of the ring
$K_{0}^{G}(\mathrm{Var}_{S})$ by inverting $\mathbb{L}$. In this article, we
also consider the localization $\mathscr{M}_{S,\mathrm{loc}}^{G}$ of
$\mathscr{M}_{S}^{G}$ with respect to the multiplicative family generated by
the elements $1-\mathbb{L}^{-n}$ with $n$ in $\mathbb{N}^{*}$.

Since a constructible subset $X$ of a $k$-variety is a finite disjoint union
of locally closed subsets, we can endow $X$ with good $G$-action via its
locally closed subsets. A constructible morphism is $G$-equivariant if its
graph admits a good $G$-action induced from the ones on its source and target.
So we can define categories $C\mathrm{Var}_{S}^{G}$ and
$\mathrm{Cons}_{S}^{G}$ as follows. As avove, fix a good $G$-action on the
$k$-variety $S$. Objects of $C\mathrm{Var}_{S}^{G}$ are $G$-equivariant
constructible morphisms from constructible sets endowed with a good $G$-action
to $S$ (over $k$), the set of morphisms between objects $X\to S$ and $Y\to S$
is

$\mathrm{Mor}_{C\mathrm{Var}_{S}^{G}}(X\to S,Y\to
S):=K_{0}(\mathrm{Var}_{X\times_{S}Y}^{G}).$

Objects of $\mathrm{Cons}_{S}^{G}$ are objects of $C\mathrm{Var}_{S}^{G}$, and
a morphism of $\mathrm{Cons}_{S}^{G}$ from $X\to S$ to $Y\to S$ is

$\left[(\mathrm{Id},f):X\to X\times_{S}Y\right],$

where $f:X\to Y$ is a $G$-equivariant constructible $S$-morphism of
constructible sets. Similarly as previous, we can define the $G$-equivariant
Grothendieck ring $K_{0}(\mathrm{Cons}_{S}^{G})$ in the usual way, and obtain
a canonical isomorphism of rings

$K_{0}^{G}(\mathrm{Var}_{S})\cong K_{0}(\mathrm{Cons}_{S}^{G}).$

Let $\hat{\mu}$ be the group scheme of roots of unity, which is the projective
limit of group schemes $\mu_{n}=\mathrm{Spec}k[t]/(t^{n}-1)$ together
transitions $\mu_{mn}\to\mu_{n}$ induced by $\lambda\mapsto\lambda^{m}$. A
good $\hat{\mu}$-action on an $S$-variety $X$ is a good $\mu_{n}$-action on
the $S$-variety $X$ for some $n$ in $\mathbb{N}^{*}$. We define

$K_{0}^{\hat{\mu}}(\mathrm{Var}_{S})=\varinjlim
K_{0}^{\mu_{n}}(\mathrm{Var}_{S}),\quad\mathscr{M}_{S}^{\hat{\mu}}=K_{0}^{\hat{\mu}}(\mathrm{Var}_{S})\left[\mathbb{L}^{-1}\right],$

and

$\mathscr{M}_{S,\mathrm{loc}}^{\hat{\mu}}=K_{0}^{\hat{\mu}}(\mathrm{Var}_{S})\left[\mathbb{L}^{-1},(\mathbb{L}^{n}-1)^{-1}\right]_{n\in\mathbb{N}^{*}}.$

Clearly, we have the identities

$\mathscr{M}_{S}^{\hat{\mu}}=\varinjlim\mathscr{M}_{S}^{\mu_{n}}\quad\text{and}\quad\mathscr{M}_{S,\mathrm{loc}}^{\hat{\mu}}=\varinjlim\mathscr{M}_{S,\mathrm{loc}}^{\mu_{n}}.$

By abuse of notation, we shall write $\mathrm{loc}$ for any of the following
localization morphisms $\mathscr{M}_{S}\to\mathscr{M}_{S,\mathrm{loc}}$,
$\mathscr{M}_{S}^{\mu_{n}}\to\mathscr{M}_{S,\mathrm{loc}}^{\mu_{n}}$ and
$\mathscr{M}_{S}^{\hat{\mu}}\to\mathscr{M}_{S,\mathrm{loc}}^{\hat{\mu}}$.

When $S$ is $\mathrm{Spec}k$, we shall write simply $\mathrm{Var}_{k}$,
$\mathscr{M}_{k}$, $\mathscr{M}_{k}^{G}$, $\mathscr{M}_{\mathrm{loc}}$ and
$\mathscr{M}_{\mathrm{loc}}^{G}$ instead of $\mathrm{Var}_{\mathrm{Spec}k}$,
$\mathscr{M}_{\mathrm{Spec}k}$, $\mathscr{M}_{\mathrm{Spec}k}^{G}$,
$\mathscr{M}_{\mathrm{Spec}k,\mathrm{loc}}$ and
$\mathscr{M}_{\mathrm{Spec}k,\mathrm{loc}}^{G}$, respectively.

##  3\. Arc spaces and rational series

###  3.1. Arc spaces

Let $X$ be an algebraic $k$-variety. For $e\in\mathbb{N}^{*}$, let
$\mathscr{L}_{e}(X)$ be the space of $e$-jet schemes of $X$, which is actually
a $k$-scheme representing the functor sending a $k$-algebra $A$ to the set of
morphisms of $k$-schemes

$\mathrm{Spec}(A[t]/(t^{e+1}))\to X.$

Thus, the set of $A$-rational points of $\mathscr{L}_{e}(X)$ is naturally
identified with the set of $A[t]/(t^{e+1})$-rational points of $X$.

For $d\geq e$ in $\mathbb{N}^{*}$, the truncation modulo $t^{e+1}$ induces an
affine morphism of $k$-schemes

$\mathscr{L}_{d}(X)\to\mathscr{L}_{e}(X)$

denoted by $\pi_{e}^{d}$. If $X$ is a smooth variety of dimension $d$, the
morphism $\pi_{e}^{d}$ is a locally trivial fibration with fiber
$\mathbb{A}_{k}^{(d-e)\dim_{k}X}$.

The above jet schemes $\mathscr{L}_{e}(X)$ and truncation morphisms
$\pi_{e}^{d}$ form in a natural way a projective system of $k$-schemes. As the
truncation morphisms are affine, the projective limit of this system exists in
the category of $k$-schemes and is called _arc space of $X$_ and denoted by
$\mathscr{L}(X)$ with truncation morphisms

$\pi_{e}:\mathscr{L}(X)\rightarrow\mathscr{L}_{e}(X).$

If $k\subseteq K$ is a field extension of $k$, then the $K$-rational points of
$\mathscr{L}(X)$ correspond one-to-one to the $K[\\![t]\\!]$-rational points
of $X$.

Recall from [8, Section 2], for any algebraically closed field $K$ containing
$k$, that a subset of $K(\\!(t)\\!)^{m}\times\mathbb{Z}^{r}$ is semi-algebraic
if it is a finite boolean combination of sets of the forms

(3.1) $\displaystyle\left\\{(x,\alpha)\in
K(\\!(t)\\!)^{m}\times\mathbb{Z}^{r}\mid\mathrm{ord}_{t}f(x)\geq\mathrm{ord}_{t}g(x)+\ell(\alpha)\right\\},$

and

(3.2) $\displaystyle\left\\{(x,\alpha)\in
K(\\!(t)\\!)^{m}\times\mathbb{Z}^{r}\mid\mathrm{ord}_{t}f(x)\equiv\ell(\alpha)\mod
n\right\\},$

and

(3.3) $\displaystyle\left\\{(x,\alpha)\in
K(\\!(t)\\!)^{m}\times\mathbb{Z}^{r}\mid\Phi(\overline{\mathrm{ac}}(f_{1}(x)),\dots,\overline{\mathrm{ac}}(f_{p}(x)))=0\right\\},$

where $f$, $g$, $f_{i}$ and $\Phi$ are $k$-polynomials, $\ell$ is a
$\mathbb{Z}$-polynomial of degree at most 1, $n$ is in $\mathbb{N}$, and
$\overline{\mathrm{ac}}(f_{i}(x))$ is the angular component of $f_{i}(x)$. One
calls a collection of formulas defining a semi-algebraic set a semi-algebraic
condition. A subset $A$ of $\mathscr{L}(X)$ is called semi-algebraic if there
exists a covering of $X$ by affine Zariski open sets $U$ such that
$A\cap\mathscr{L}(U)$ is of the form

(3.4) $\displaystyle
A\cap\mathscr{L}(U)=\\{x\in\mathscr{L}(U)\mid\theta(f_{1}(\tilde{x}),\dots,f_{p}(\tilde{x});\alpha)\\},$

where $f_{i}$ are regular functions on $U$, $\theta$ is a semi-algebraic
condition, $\alpha$ may be a given tuple of integers or nothing, and
$\tilde{x}$ is the element in $\mathscr{L}(U)(k(x))$ corresponding to a point
$x$ in $\mathscr{L}(U)$ of residue field $k(x)$.

By [17], if $g:X\to Y$ is a morphism of algebraic $k$-varieties and $A$ is a
semi-algebraic subset of $\mathscr{L}(X)$, then $g(A)$ is a semi-algebraic
subset of $Y$. Then the map $g:A\to g(A)$ is called a semi-algebraic morphism
of semi-algebraic sets. More generally, let $A$ and $B$ be semi-algebraic
subsets of $\mathscr{L}(X)$ and $\mathscr{L}(Y)$, the arc spaces of
$k$-varieties $X$ and $Y$, respectively, and let $h:A\to B$ be a map. Then $h$
is called a semi-algebraic morphism if its graph is a semi-algebraic subset of
$\mathscr{L}(X\times_{k}Y)$. Denote by $\mathrm{SA}_{k}$ be the category whose
objects are pairs $(A,\mathscr{L}(X))$, where $A$ is a semi-algebraic subset
of the arc space $\mathscr{L}(X)$ of an algebraic $k$-variety $X$, and a
morphism of $\mathrm{SA}_{k}$ between two objects $(A,\mathscr{L}(X))$ and
$(B,\mathscr{L}(Y))$ is a semi-algebraic morphism of semi-algebraic sets $A\to
B$. For a given $k$-variety $X$, we can consider the full subcategory
$\mathrm{SA}_{k}(X)$ of $\mathrm{SA}_{k}$ consists of semi-algebraic subsets
of $\mathscr{L}(X)$.

In the sense of [8, Definition-Proposition 3.2], Denef-Loeser’s motivic volume
is defined on $\mathrm{Ob}\mathrm{SA}_{k}(X)$ the set of all the semi-
algebraic subsets of $\mathscr{L}(X)$ with the reasonable properties. By [4,
Remark 16.3.2], this motivic volume essentially takes values in
$\mathscr{M}_{\mathrm{loc}}$. In the present article, we denote Denef-Loeser’s
motivic volume by $\mu^{\prime}$ (the symbol $\mu$ will be devoted to indicate
Cluckers-Loeser’s motivic volume [4]).

###  3.2. Rationality

Let $\mathscr{M}$ be a commutative ring with unity containing $\mathbb{L}$ and
$\mathbb{L}^{-1}$, and let $\mathscr{M}[[T]]$ be the set of formal power
series in $T$ with coefficients in $\mathscr{M}$, which is a ring and also a
$\mathscr{M}$-module with respect to usual operations for series. Denote by
$\mathscr{M}[[T]]_{\mathrm{sr}}$ the submodule of $\mathscr{M}[[T]]$ generated
by 1 and by finite products of terms

$\frac{\mathbb{L}^{p}T^{q}}{(1-\mathbb{L}^{p}T^{q})}$

for $(p,q)$ in $\mathbb{Z}\times\mathbb{N}^{*}$. An element of
$\mathscr{M}[[T]]_{\mathrm{sr}}$ is called a rational series. By [7], there
exists a unique $\mathscr{M}$-linear morphism

$\lim_{T\to\infty}:\mathscr{M}[[T]]_{\mathrm{sr}}\to\mathscr{M}$

such that for any $(p,q)$ in $\mathbb{Z}\times\mathbb{N}^{*}$,

$\lim_{T\to\infty}\frac{\mathbb{L}^{p}T^{q}}{(1-\mathbb{L}^{p}T^{q})}=-1.$

Let us recall some examples on the rationality. Let $X$ be a smooth algebraic
$k$-variety of pure dimension $m$, and $f$ a regular function on $X$ with zero
locus $X_{0}\not=\emptyset$. For $e$ in $\mathbb{N}^{*}$, put

$X[e]=\left\\{\gamma\in\mathscr{L}_{e}(X)\mid f(\gamma)=t^{e}\mod
t^{e+1}\right\\},$

which is naturally an $X_{0}$-variety and stable under the action
$\lambda\cdot\gamma(t):=\gamma(\lambda t)$ of $\mu_{e}$ on
$\mathscr{L}_{e}(X)$. Write simply $\big{[}X[e]\big{]}$ for the class
$\big{[}X[e]\to X_{0}\big{]}$ in $\mathscr{M}_{X_{0}}^{\mu_{e}}$. It is proved
in [7] that the series

$\displaystyle
Z_{f}(T):=\sum_{e\in\mathbb{N}^{*}}\big{[}X[e]\big{]}\mathbb{L}^{-em}T^{e},$

is a rational series, i.e., in
$\mathscr{M}_{X_{0}}^{\hat{\mu}}[[T]]_{\mathrm{sr}}$. More general, we can
obtain the rationality of a series generalizing $Z_{f}(T)$ without assuming
that $X$ is smooth, and with $f$ concerning several semi-algebraic subsets in
$\mathscr{L}(X)$. Let $\mu^{\prime}$ be Denef-Loeser’s motivic volume defined
in [8]. The below theorem is a result given in [13, Proposition 4.6].

######  Theorem 3.1 (Lê-Nguyen [13]).

Let $X$ be a $k$-variety and $f$ a regular function on $X$. Let $A_{\alpha}$,
$\alpha$ in $\mathbb{N}^{r}$, be a family of semi-algebraic subsets of
$\mathscr{L}(X)$ such that there exists a covering of $X$ by affine Zariski
open sets $U$ satisfying the condition that $A_{\alpha}\cap\mathscr{L}(U)$ are
finite boolean combinations of sets of the forms (3.1) and (3.2). Assume that,
for every $\alpha$ in $\mathbb{N}^{r}$, $A_{\alpha}$ is stable in the sense of
[8] and disjoint with $\mathscr{L}(X_{\text{Sing}})$. For
$e\in\mathbb{N}^{*}$, we put

$A_{e,\alpha}:=\left\\{\gamma\in A_{\alpha}\mid f(\gamma)=t^{e}\mod
t^{e+1}\right\\}.$

Let $\Delta$ be a rational polyhedral convex cone in $\mathbb{R}^{r+1}_{\geq
0}$ and $\bar{\Delta}$ its closure. Let $\ell$ and $\ell^{\prime}$ be integral
linear forms on $\mathbb{Z}^{r+1}$ with $\ell(e,\alpha)>0$ and
$\ell^{\prime}(e,\alpha)\geq 0$ for all $(e,\alpha)$ in
$\bar{\Delta}\setminus\\{0\\}$. Then the formal power series

$Z(T):=\sum_{(e,\alpha)\in\Delta\cap\mathbb{N}^{r+1}}\mu^{\prime}\left(A_{e,\alpha}\right)\mathbb{L}^{-\ell^{\prime}(e,\alpha)}T^{\ell(e,\alpha)}$

is an element of $\mathscr{M}_{k}^{\hat{\mu}}[[T]]_{\mathrm{sr}}$, and the
limit $\lim_{T\to\infty}Z(T)$ is independent of such an $\ell$ and
$\ell^{\prime}$.

##  4\. Categories of definable subassignments

In this section, we recall some concepts and results on motivic integration in
the sense of Cluckers-Loeser [4]. We also provide an equivariant version for
objects concerning definable $T_{\mathrm{acl}}$-subassignments, where
$T_{\mathrm{acl}}$ is the theory of all algebraically closed fields containing
$k$.

###  4.1. Definable subassignments

We consider the formalism of Cluckers and Loeser [4] with a concrete Denef-Pas
language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}$ consisting of the ring
language $\mathbf{L}_{\mathrm{Rings}}=\left\\{+,-,\cdot,0,1\right\\}$ for
valued fields, also the ring language $\mathbf{L}_{\mathrm{Rings}}$ for
residue fields, and the Presburger language $\mathbf{L}_{\mathrm{PR}}$ for
value groups, where

$\mathbf{L}_{\mathrm{PR}}=\left\\{+,-,0,1,\leq\right\\}\cup\left\\{\equiv_{n}\mid
n\in\mathbb{N}^{*}\right\\},$

and $\equiv_{n}$ is the equivalence relation modulo $n$.

Let $\mathrm{Field}_{k}$ be the category of all fields $K$ containing $k$
whose morphisms are field morphisms. For any $K$ in $\mathrm{Field}_{k}$, we
consider the natural valuation map
$\mathrm{ord}_{t}:K(\\!(t)\\!)^{\times}\to\mathbb{Z}$ augmented by
$\mathrm{ord}_{t}(0)=+\infty$, and the natural angular component map
$\overline{\mathrm{ac}}:K(\\!(t)\\!)^{\times}\to K$ augmented by
$\overline{\mathrm{ac}}(0)=0$.

For a basic set

$V:=\mathbb{A}_{k(\\!(t)\\!)}^{m}\times\mathbb{A}_{k}^{n}\times\mathbb{Z}^{r},$

with $m$, $n$, $r$ in $\mathbb{N}$, we consider the functor $h_{V}$ (also
denoted by $h[m,n,r]$) from $\mathrm{Field}_{k}$ to the category of sets
defined by

$h_{V}(K)=h[m,n,r](K):=K(\\!(t)\\!)^{m}\times K^{n}\times\mathbb{Z}^{r}.$

If $\mathsf{X}$ is a map sending each object $K$ of $\mathrm{Field}_{k}$ to a
subset $\mathsf{X}(K)$ of $K(\\!(t)\\!)^{m}\times K^{n}\times\mathbb{Z}^{r}$,
then $\mathsf{X}$ is called a affine subassignment (or, shortly speaking,
subassignment) of $h_{V}=h[m,n,r]$. Note that $\mathsf{X}$ is not necessarily
a subfunctor of $h[m,n,r]$. In the same way, we can define morphisms of
subassignments and their graph, as well as union, subtraction, Cartesian
product and fiber product of two subassignments.

A subassignment $\mathsf{X}$ of $h[m,n,r]$ is called definable if there exists
a formula $\varphi$ in $\mathcal{L}_{\mathrm{DP},\mathrm{P}}$ with
$k(\\!(t)\\!)$-coefficients and $m$ free variables in the valued field sort,
$k$-coefficients and $n$ free variables in residue field sort, and $r$ free
variables in the value group sort, such that, for any $K$ in
$\mathrm{Field}_{k}$,

$\mathsf{X}(K)=\left\\{x\in K(\\!(t)\\!)^{m}\times
K^{n}\times\mathbb{Z}^{r}\mid(K(\\!(t)\\!),K,\mathbb{Z})\models\varphi(x)\right\\}.$

In this setting, we also write $h_{\varphi}$ for the definable subassignment
$\mathsf{X}$. Denote by $\emptyset$ the empty definable subassignment, with
$\emptyset(K)=\emptyset$ for any $K$ in $\mathrm{Field}_{k}$. For $\mathsf{X}$
and $\mathsf{X}^{\prime}$ being definable subassignments of $h[m,n,r]$ and
$h[m^{\prime},n^{\prime},r^{\prime}]$, respectively, a definable morphism
$\mathsf{X}\to\mathsf{X}^{\prime}$ is a morphism of subassignments
$\mathsf{X}\to\mathsf{X}^{\prime}$ such that its graph is a definable
subassignment of $h[m+m^{\prime},n+n^{\prime},r+r^{\prime}]$.

For a set

$W:=\mathcal{X}\times X\times\mathbb{Z}^{r},$

with $\mathcal{X}$ an algebraic $k(\\!(t)\\!)$-variety, and $X$ an algebraic
$k$-variety, we define

$h_{W}(K):=\mathcal{X}(K(\\!(t)\\!))\times X(K)\times\mathbb{Z}^{r},$

for any $K$ in $\mathrm{Field}_{k}$. In general, we can define definable
subassignments of $h_{W}$, definable morphisms of definable subassignments,
and usual operations on definable subassignments of functors of the form
$h_{W}$ using a glueing procedure (as in [4, Section 2.3]): taking finite
covers (always exist) of $\mathcal{X}$ and $X$ by affine open
$k(\\!(t)\\!)$-subvarieties and $k$-subvarieties, respectively, going back to
the definition of affine definable subassignment, and glueing them.

We consider the category
$\mathrm{Def}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}})$ (or
$\mathrm{Def}_{k}$ for short) of affine definable subassignments, where its
objects are pairs $(\mathsf{X},h[m,n,r])$ with $\mathsf{X}$ being a definable
subassignment of $h[m,n,r]$, and a morphism

$(\mathsf{X},h[m,n,r])\to(\mathsf{X}^{\prime},h[m^{\prime},n^{\prime},r^{\prime}])$

in $\mathrm{Def}_{k}$ is a definable morphism
$\mathsf{X}\to\mathsf{X}^{\prime}$. We also consider the category
$\mathrm{GDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}})$ (or
$\mathrm{GDef}_{k}$ for short) of global definable subassignments, where
objects of $\mathrm{GDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}})$ are pairs
$(\mathsf{X},h_{W})$ with $h_{W}$ as above and $\mathsf{X}$ being a definable
subassignment of $h_{W}$, and a morphism

$(\mathsf{X},h_{W})\to(\mathsf{X}^{\prime},h_{W^{\prime}})$

in $\mathrm{GDef}_{k}$ is a definable morphism
$\mathsf{X}\to\mathsf{X}^{\prime}$. For any affine definable subassignment
$\mathsf{S}$, we denote by
$\mathrm{Def}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}})$ (or
$\mathrm{Def}_{\mathsf{S}}$ for short) the category of morphisms
$\mathsf{X}\to\mathsf{S}$ in $\mathrm{Def}_{k}$, and a morphism in
$\mathrm{Def}_{\mathsf{S}}$ between $\mathsf{X}\to\mathsf{S}$ and
$\mathsf{X}^{\prime}\to\mathsf{S}$ is a morphism
$\mathsf{X}\to\mathsf{X}^{\prime}$ in $\mathrm{Def}_{k}$ which is compatible
with the morphisms to $\mathsf{S}$. For any definable subassignment
$\mathsf{S}$, the category
$\mathrm{GDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}})$ (or
$\mathrm{GDef}_{\mathsf{S}}$ for short) can be defined in the same way as
$\mathrm{GDef}_{k}$ with $h_{\mathrm{Spec}k}$ replaced by $\mathsf{S}$.

When we consider the category $\mathrm{Field}_{k}(T_{\mathrm{acl}})$ of all
algebraically closed fields containing $k$ in stead of $\mathrm{Field}_{k}$,
and use the language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$ (see the
definition in Section 4.3) instead of $\mathcal{L}_{\mathrm{DP},\mathrm{P}}$,
we obtain corresponding notions of $T_{\mathrm{acl}}$-subassignment,
$\mathrm{Def}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and
$\mathrm{GDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$.

###  4.2. Points on definable subassignments

Let $\mathsf{X}$ be an object in $\mathrm{GDef}_{k}$. A point $x$ on
$\mathsf{X}$ is a tuple $x=(x_{0},K)$ such that $K$ is in $\mathrm{Field}_{k}$
and $x_{0}$ is in $\mathsf{X}(K)$. For such a point $x$ on $\mathsf{X}$ we
usually write $k(x)$ for $K$ and call it the residue field of $x$. Let

$f:\mathsf{X}\to\mathsf{Y}$

be a morphism in $\mathrm{Def}_{k}$, with

$\mathsf{X}=(\mathsf{X}_{0},h[m,n,r])$

and

$\mathsf{Y}=(\mathsf{Y}_{0},h[m^{\prime},n^{\prime},r^{\prime}]),$

whose graph is defined by a formula $\varphi(x,y)$, where $x$ is in $h[m,n,r]$
and $y$ is in $h[m^{\prime},n^{\prime},r^{\prime}]$. One defines the fiber of
$f$ over a point $y=(y_{0},k(y))$ on $\mathsf{Y}$ to be the definable
subsassignment $\mathsf{X}_{y}$ in $\mathrm{Def}_{k(y)}$ given by the formula
$\varphi(x,y_{0})$. In the category $\mathrm{GDef}_{k}$, fibers of a morphism
are defined in the same way by using affine covers.

###  4.3. Categories
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
$\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$

Denote by $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$ the language extending
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}$ by adding constants in $k$ so that all
polynomials in both valued field sort and residue field sort having
coefficients in $k$. Let $X$ be an algebraic $k$-variety, let

$\mathcal{X}:=X\times_{k}\mathrm{Spec}k(\\!(t)\\!),$

and $\mathsf{A}$ a definable subassignment of $h_{\mathcal{X}}$ defined by a
formula in $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$. Assume first
$\mathcal{X}$ is a closed subscheme in $\mathbb{A}_{k(\\!(t)\\!)}^{m}$, for
some $m$ in $\mathbb{N}$, such that the ideal defining $\mathcal{X}$ generated
by polynomials with coefficients in $k[[t]]$. The above mentioned definable
subassignment $\mathsf{A}$ (i.e., defined by a formula in
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$) is called small if $\mathsf{A}$ is
contained in the following definable subassignment

$\left\\{(x_{1},\dots,x_{m})\in h[m,0,0]\mid\mathrm{ord}_{t}x_{i}\geq 0,1\leq
i\leq m\right\\}.$

For $\mathcal{X}$ not necessarily affine, we call $\mathsf{A}$ small if there
exists a cover of $\mathcal{X}$ by open affine $k(\\!(t)\\!)$-subvarieties
$\mathcal{U}_{i}$ defined by the vanishing of polynomials with coefficients in
$k[[t]]$ such that $\mathsf{A}\cap h_{\mathcal{U}_{i}}$ are small for all $i$.
Let
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
be the subcategory of
$\mathrm{GDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}},T_{\mathrm{acl}})$
whose objects are pairs

$(\mathsf{A},h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)}),$

where $X$ is an algebraic $k$-variety and $\mathsf{A}$ is a small definable
$T_{\mathrm{acl}}$-subassignment of
$h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)}$, and a morphism in
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
between objects

$(\mathsf{A},h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)})$

and

$(\mathsf{B},h_{Y\times_{k}\mathrm{Spec}k(\\!(t)\\!)})$

is a $T_{\mathrm{acl}}$-morphism of $T_{\mathrm{acl}}$-subassignments
$\mathsf{A}\to\mathsf{B}$ such that its graph is a small definable
$T_{\mathrm{acl}}$-subassignment of
$h_{X\times_{k}Y\times_{k}\mathrm{Spec}k(\\!(t)\\!)}$.

Fixing an algebraic $k$-variety $X$, we define a category denoted by
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
which is the full subcategory of
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
whose objects contain all the small definable
$T_{\mathrm{acl}}$-subassignments of
$h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)}$.

######  Theorem 4.1.

The categories
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{SA}_{k}$ are equivalent. If $X$ is an algebraic $k$-variety, then
the categories
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{SA}_{k}(X)$ are equivalent.

###### Proof.

At the moment, for short, we write $(\mathsf{A},X)$ instead of
$(\mathsf{A},h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)})$ for an object of
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
and $(A,X)$ instead of $(A,\mathscr{L}(X))$ for an object in of
$\mathrm{SA}_{k}$.

First, let us construct a functor $\mathcal{F}$ from
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
to $\mathrm{SA}_{k}$. Let $(\mathsf{A},X)$ be an object of
$\mathrm{SDef}_{k}(X,\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$.
Then, there is a cover of $X\times_{k}\mathrm{Spec}k(\\!(t)\\!)$ by Zariski
open affine $k(\\!(t)\\!)$-subvarieties $\mathcal{U}$, with $\mathcal{U}$
embedded as a closed $k(\\!(t)\\!)$-subvariety in some
$\mathbb{A}_{k(\\!(t)\\!)}^{m}$ (we can take $m$ common for all $\mathcal{U}$)
and the embedding defined over $k[[t]]$, such that for the standard
coordinates $x_{i}$ of $h[m,0,0]$ and any point $x$ on $\mathsf{A}\cap
h_{\mathcal{U}}$ we have $\mathrm{ord}_{t}x_{i}(x)\geq 0$. Moreover,
$\mathsf{A}\cap h_{\mathcal{U}}$ is defined by a formula
$\varphi(\underline{x},\alpha)$ in the language
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$, where
$\underline{x}=(x_{1},\dots,x_{m})$ and $\alpha=(\alpha_{1},\dots,\alpha_{r})$
are free variables in value group sort, namely,

$\mathsf{A}\cap h_{\mathcal{U}}=\left\\{x\in
h_{\mathcal{U}}\mid\varphi(x_{1}(x),\dots,x_{m}(x),\alpha)\right\\}.$

By Denef-Pas’ quantifier elimination for algebraically closed fields (cf.
Corollary 2.1.2 of [4]), $\varphi(\underline{x},\alpha)$ is equivalent to a
finite disjunction of formulas of the form

(4.1)
$\displaystyle\psi(\overline{\mathrm{ac}}g_{1}(\underline{x}),\dots,\overline{\mathrm{ac}}g_{q}(\underline{x}))\wedge\vartheta(\mathrm{ord}_{t}f_{1}(\underline{x}),\dots,\mathrm{ord}_{t}f_{p}(\underline{x}),\alpha),$

where $f_{i}$ and $g_{j}$ are polynomials over $k$, $\psi$ is an
$\mathbf{L}_{\mathrm{Rings}}$-formula with coefficients in $k$, and
$\vartheta$ is an $\mathbf{L}_{\mathrm{PR}}$-formula. Thus, as seen in (3.1),
(3.2) and (3.3), the formula $\varphi$ is nothing but a semi-algebraic
condition.

Since all polynomials $f_{i}$ and $g_{j}$ have coefficients in $k$, in
particular, polynomials defining $\mathcal{U}$ have coefficients in $k$, there
exists a unique closed $k$-subvariety $U$ in $\mathbb{A}_{k}^{m}$ such that

(4.2) $\displaystyle\mathcal{U}=U\times_{k}\mathrm{Spec}k(\\!(t)\\!),$

hence we have a cover $\\{U\\}_{U}$ of $X$ by Zariski open affine
$k$-subvarieties. The above $x_{i}$ induce regular functions $x^{\prime}_{i}$
on $U$ such that $x^{\prime}_{i}$ are standard coordinate components in
$\mathbb{A}_{k}^{m}$ for every $1\leq i\leq n$, and that $U$ is defined by the
vanishing of $x^{\prime}_{i}$ for $n+1\leq i\leq m$. Now we put

$A_{U}:=\left\\{x\in\mathscr{L}(U)\mid\varphi(x^{\prime}_{1}(\tilde{x}),\dots,x^{\prime}_{m}(\tilde{x}),\alpha)\right\\},$

where $\tilde{x}$ is defined after (3.4), and glue $A_{U}$’s into a semi-
algebraic subset $A$ of $\mathscr{L}(X)$. Note that the construction of $A$ is
up to semi-algebraic isomorphism independent of the choice of the cover
$\\{\mathcal{U}\\}_{\mathcal{U}}$. Let us define

$\mathcal{F}(\mathsf{A},X):=(A,X),$

which is an object in $\mathrm{SA}_{k}$.

We shall construct a morphism $\mathcal{F}(\mathsf{f})$ in $\mathrm{SA}_{k}$
which is the image under $\mathcal{F}$ of a morphism $\mathsf{f}$ in
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
such that

(4.3)
$\displaystyle\mathcal{F}(\mathsf{g}\circ\mathsf{f})=\mathcal{F}(\mathsf{g})\circ\mathcal{F}(\mathsf{f}).$

Consider a morphism $\mathsf{f}:(\mathsf{A},X)\to(\mathsf{B},Y)$, written
$\mathsf{f}:\mathsf{A}\to\mathsf{B}$ for short, in
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$.
Then we can cover $X\times_{k}k(\\!(t)\\!)$ (resp. $Y\times_{k}k(\\!(t)\\!))$
by Zariski open affine $k(\\!(t)\\!)$-subvarieties $\mathcal{U}$ (resp.
$\mathcal{U}^{\prime}$) such that $\mathsf{A}\cap h_{\mathcal{U}}$ and
$\mathsf{B}\cap h_{\mathcal{U}^{\prime}}$ are defined by formulas, say
$\psi_{\mathcal{U}}$ and $\phi_{\mathcal{U}^{\prime}}$, respectively, which
are disjunction of formulas of the form (4.1). We use the covering
$\\{\mathcal{U}\\}$ of $X\times_{k}k(\\!(t)\\!)$ and the formulas
$\psi_{\mathcal{U}}$ (resp. the covering $\\{\mathcal{U}^{\prime}\\}$ of
$Y\times_{k}k(\\!(t)\\!)$ and the formulas $\phi_{\mathcal{U}^{\prime}}$) to
construct a constructible subset $\mathcal{A}$ (resp. $\mathcal{B}$) of
$X\times_{k}k(\\!(t)\\!)$ (resp. $Y\times_{k}k(\\!(t)\\!)$). The morphism
$\mathsf{f}$ induces a morphism of constructible sets
$\mathcal{A}\to\mathcal{B}$, hence with the same reason as the existence of
$U$ in (4.2) the morphism $\mathcal{A}\to\mathcal{B}$ in its turn induces a
semi-algebraic morphism of semi-algebraic sets $f:(A,X)\to(B,Y)$. So we define

$\mathcal{F}(\mathsf{f}):=f,$

which is well defined and a morphism in $\mathrm{SA}_{k}$.

Since any morphism $\mathsf{f}$ can be factorized through the inclusion into
its graph followed by a projection, to check the preserving property (4.3) it
suffices to do for inclusions and projections of small definable
$T_{\mathrm{acl}}$-subassignments. By definition, for the
$\mathsf{B}$-projection

$\mathsf{pr}_{\mathsf{B}}:(\mathsf{A}\times\mathsf{B},X\times_{k}Y)\to(\mathsf{B},Y)$

we have

$\mathcal{F}(\mathsf{pr}_{\mathsf{B}})=\mathrm{pr}_{B}:A\times B\to B,$

the $B$-projection of semi-algebraic subset $A\times B$ of
$\mathscr{L}(X\times_{k}Y)=\mathscr{L}(X)\times_{k}\mathscr{L}(Y)$ onto $B$,
which is a morphism in $\mathrm{SA}_{k}$. Also, for a morphism
$\mathsf{i}_{\mathsf{AB}}:(\mathsf{A},X)\to(\mathsf{B},X)$ in
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
induced by an inclusion $\mathsf{A}\hookrightarrow\mathsf{B}$ of small
definable $T_{\mathrm{acl}}$-subassignments, we have
$\mathcal{F}(\mathsf{i}_{\mathsf{AB}})(x)=x$ for all $x\in A$. Now, let us
consider

$\mathsf{i}:(\mathsf{C},X\times_{k}Y)\to(\mathsf{A}\times\mathsf{B},X\times_{k}Y),$

which is a morphism in the category
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
induced by an inclusion $\mathsf{C}\hookrightarrow\mathsf{A}\times\mathsf{B}$.
By definition, it is clear that

$\mathcal{F}(\mathsf{pr}_{\mathsf{B}}\circ\mathsf{i})=\mathcal{F}(\mathsf{pr}_{\mathsf{B}})\circ\mathcal{F}(\mathsf{i}).$

In the same way, for a morphism

$\mathsf{j}:(\mathsf{B},Y)\to(\mathsf{E},Y)$

in $\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k))$ induced by an
inclusion $\mathsf{B}\hookrightarrow\mathsf{E}$, we have

$\mathcal{F}(\mathsf{j}\circ\mathsf{pr}_{\mathsf{B}})=\mathcal{F}(\mathsf{j})\circ\mathcal{F}(\mathsf{pr}_{\mathsf{B}}).$

We now construct a functor $\mathcal{G}$ from $\mathrm{SA}_{k}$ to
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
which is naturally inverse to $\mathcal{F}$. Let $(A,X)$ be an object of
$\mathrm{SA}_{k}$, i.e., $A$ is a semi-algebraic subset of $\mathscr{L}(X)$.
By definition, there exist a cover of $X$ by Zariski open affine
$k$-subvarieties $V$ (viewed as a closed $k$-subvariety of
$\mathbb{A}_{k}^{m}$), and for each $V$, regular functions $h_{i}$ on $V$,
$1\leq i\leq m$, a semi-algebraic condition $\varphi$ (with $h_{i}$ and
$\varphi$ depending on $V$) such that

$A\cap\mathscr{L}(V)=\left\\{x\in\mathscr{L}(V)\mid\varphi(h_{1}(\tilde{x}),\dots,h_{m}(\tilde{x}),\alpha)\right\\}.$

Clearly,

$\mathcal{V}:=V\times_{k}\mathrm{Spec}k(\\!(t)\\!)$

is embedded over $k[[t]]$ into $\mathbb{A}_{k(\\!(t)\\!)}^{m}$, and they form
a cover of $X\times_{k}\mathrm{Spec}k(\\!(t)\\!)$. Note that $\varphi$ is a
formula in the language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$, and that
each $h_{i}$ induces a definable morphism of definable subassignments

$x_{i}:h_{\mathcal{V}}\to h[1,0,0].$

Put

$\mathsf{A}_{\mathcal{V}}=\left\\{x\in
h_{\mathcal{V}}\mid\varphi(x_{1}(x),\dots,x_{m}(x),\alpha),\mathrm{ord}_{t}x_{i}(x)\geq
0,1\leq i\leq m\right\\},$

and glue all $\mathsf{A}_{\mathcal{V}}$ along the cover $\\{\mathcal{V}\\}$ of
$X\times_{k}\mathrm{Spec}k(\\!(t)\\!)$ to get a small definable
$T_{\mathrm{acl}}$-subassignment $\mathsf{A}$ of
$h_{X\times_{k}\mathrm{Spec}k(\\!(t)\\!)}$. We can prove that the construction
of $\mathsf{A}$ is up to definable isomorphism independent of the choice of
the cover $\\{V\\}$. So we can define

$\mathcal{G}(A,X)=(\mathsf{A},X),$

which is an object of
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$.
Similarly, we can define $\mathcal{G}(f)$ to be a morphism of
$\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
when $f$ is a morphism of $\mathrm{SA}_{k}$, which satisfies

$\mathcal{G}(f\circ g)=\mathcal{G}(f)\circ\mathcal{G}(g).$

The existence of natural isomorphisms

$\varepsilon:\mathcal{F}\circ\mathcal{G}\to\mathrm{Id}_{\mathrm{SA}_{k}}$

and

$\eta:\mathrm{Id}_{\mathrm{SDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})}\to\mathcal{G}\circ\mathcal{F}$

follows from the fact that the construction of $\mathsf{A}$ from $A$ and vice
versa is independent of the choice of covers by open affine subvarieties. ∎

Let $\mathsf{S}$ be an object in
$\mathrm{GDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$.
Let
$\mathrm{RDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
denote the full subcategory of
$\mathrm{GDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
such that each object $\mathsf{X}\to\mathsf{S}$ of
$\mathrm{RDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
is the $\mathsf{S}$-projection of a definable subassignment $\mathsf{X}$ of
$\mathsf{S}\times h_{\mathbb{A}_{k}^{n}}$, for some $n$ in $\mathbb{N}$. We
first mention a special case when $\mathsf{S}=h_{S}$ as follows. Let $S$ be a
closed $k$-subvariety of $\mathbb{A}_{k}^{d}$, for a given $d$ in
$\mathbb{N}$. Then the category
$\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
defined previously is just the full subcategory of
$\mathrm{Def}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
its objects $\mathsf{X}\to h_{S}$ are the $h_{S}$-projection of definable
subassignments $\mathsf{X}$ of $h_{S\times_{k}\mathbb{A}_{k}^{n}}$, with $n$
being variable in $\mathbb{N}$.

######  Theorem 4.2.

For any $k$-variety $S$, the categories
$\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{Cons}_{S}$ are equivalent.

###### Proof.

Using the argument in [4, Section 16.2], under some elimination theorems, we
have that objects of
$\mathrm{Def}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
are defined by formulas without quantifiers in the Denef-Pas language
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}$. Since Chevalley’s constructibility
theorem in algebraic geometry (over $k$) is nothing else than the quantifier
elimination theorem for the theory of algebraically closed fields containing
$k$, a formula defining a definable subassignment of
$h_{S\times_{k}\mathbb{A}_{k}^{n}}$ defines a constructible subset of
$S\times_{k}\mathbb{A}_{k}^{n}$, and via graph, a definable morphism of
definable subassignments gives rise to a constructible morphism of
constructible sets, and vice versa. For a detailed argument, we can use the
strategy in the proof of Theorem 4.1. ∎

###  4.4. Actions

Let $X$ be an algebraic $k$-variety, and $G$ an algebraic group over $k$. A
$G$-action (or $h_{G}$-action) on $h_{X}$ is a definable morphism of definable
subassignments

$h_{G\times_{k}X}\to h_{X}$

such that the corresponding morphism of $k$-varieties

$G\times_{k}X\to X$

is a $G$-action on $X$. The $G$-action on $h_{X}$ is called good if the
corresponding $G$-action on $X$ is good. In this setting, a definable morphism
of definable subassignments

$h_{X}\to h_{Y}$

is $G$-equivariant if the corresponding morphism of $k$-varieties

$X\to Y$

is $G$-equivariant. By Theorem 4.2, we can extend this definition of good
$G$-action to that on any definable subassignment of $h[0,n,0]$, for $n$ in
$\mathbb{N}$. Let $S$ be a closed $k$-subvariety of $\mathbb{A}_{k}^{d}$, and
let $S$ be endowed with a good $G$-action. Denote by
$\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
the subcategory of
$\mathrm{GDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
whose objects are $G$-equivariant definable $T_{\mathrm{acl}}$-morphisms of
definable $T_{\mathrm{acl}}$-subassignments $\mathsf{X}\to h_{S}$, where
$\mathsf{X}$ is a definable $T_{\mathrm{acl}}$-subassignment of
$h_{S\times_{k}\mathbb{A}_{k}^{n}}$, for some $n$ in $\mathbb{N}$, and
$\mathsf{X}$ is endowed with a good $G$-action, a morphism in
$\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
from an object $\mathsf{X}\to h_{S}$ to another one $\mathsf{Y}\to h_{S}$ is a
$G$-equivariant definable $T_{\mathrm{acl}}$-morphism
$\mathsf{X}\to\mathsf{Y}$ which commutes with the $G$-equivariant morphisms to
$h_{S}$.

######  Lemma 4.3.

For any $k$-variety $S$,
$\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and $\mathrm{Cons}_{S}^{G}$ are equivalent.

###### Proof.

The lemma is deduced directly from Theorem 4.2 and the definition of good
$G$-action on definable subassignments. ∎

For an algebraic $k(\\!(t)\\!)$-variety $\mathcal{X}$, the definable
subassignment $h_{\mathcal{X}}$ admits a natural $\mu_{n}$-action
$h_{\mu_{n}}\times h_{\mathcal{X}}\to h_{\mathcal{X}}$ induced by

$(\lambda,t)\mapsto\lambda t,$

for all $n$ in $\mathbb{N}^{*}$. More precisely, for every $K$ in
$\mathrm{Field}_{k}$, $\lambda$ in $\mu_{n}(K)$ and $\varphi(t)$ in
$\mathcal{X}(K(\\!(t)\\!))$, we have

$\lambda\cdot\varphi(t)=\varphi(\lambda t).$

The profinite group scheme $\hat{\mu}$ acts naturally on $h_{\mathcal{X}}$ via
$\mu_{n}$ for some $n$ in $\mathbb{N}^{*}$.

###  4.5. Grothendieck semirings and rings of definable subassignments

Let $\mathsf{S}$ be a definable subassignment. According to [4], the
Grothendieck semigroup $SK_{0}(\mathrm{RDef}_{\mathsf{S}})$ of the category
$\mathrm{RDef}_{\mathsf{S}}$ is the quotient of the free abelian semigroup
generated by symbols $[\mathsf{X}\to\mathsf{S}]$ with
$\mathsf{X}\to\mathsf{S}$ being objects in $\mathrm{RDef}_{\mathsf{S}}$ modulo
the following relations:

$[\emptyset\to\mathsf{S}]=0,$
$[\mathsf{X}\to\mathsf{S}]=[\mathsf{Y}\to\mathsf{S}]$

if $\mathsf{X}\to\mathsf{S}$ and $\mathsf{Y}\to\mathsf{S}$ are isomorphic in
$\mathrm{RDef}_{\mathsf{S}}$, and

$[\mathsf{X}\cup\mathsf{Y}\to\mathsf{S}]+[\mathsf{X}\cap\mathsf{Y}\to\mathsf{S}]=[\mathsf{X}\to\mathsf{S}]+[\mathsf{Y}\to\mathsf{S}]$

for definable subassignments $\mathsf{X}$ and $\mathsf{Y}$ of
$\mathsf{S}\times h_{\mathbb{A}_{k}^{n}}$, for some $n$ in $\mathbb{N}$, and
morphisms of $\mathsf{X}$ and $\mathsf{Y}$ to $\mathsf{S}$ factorizing through
$\mathsf{S}$-projection. Denote by $K_{0}(\mathrm{RDef}_{\mathsf{S}})$ the
group associated to the Grothendieck semigroup
$SK_{0}(\mathrm{RDef}_{\mathsf{S}})$. If we provide
$SK_{0}(\mathrm{RDef}_{\mathsf{S}})$ and $K_{0}(\mathrm{RDef}_{\mathsf{S}})$
with a product induced by the fiber product over $\mathsf{S}$ of morphisms of
subassignments to $\mathsf{S}$ defined in Section 2.2 of [4], then they are
commutative semiring and ring with unity, respectively. Remark that the
canonical morphism

$SK_{0}(\mathrm{RDef}_{\mathsf{S}})\to K_{0}(\mathrm{RDef}_{\mathsf{S}})$

is not necessarily injective.

Let $S$ be a $k$-variety endowed with a given $G$-action. The $G$-equivariant
Grothendieck group $K_{0}^{G}(\mathrm{RDef}_{h_{S}})$ is the quotient of the
free abelian group generated by symbols

$[\mathsf{X}\to h_{S},\sigma]$

with $\mathsf{X}$ being a definable subassignment of
$h_{S\times_{k}\mathbb{A}_{k}^{n}}$, for some $n$ in $\mathbb{N}$, endowed
with a good $G$-action $\sigma$, and $\mathsf{X}\to h_{S}$ being a morphism in
$\mathrm{Def}_{k}$, modulo the following relations

$[\mathsf{X}\to h_{S},\sigma]=[\mathsf{Y}\to h_{S},\sigma^{\prime}]$

if there exists a $G$-equivariant definable morphism $\mathsf{X}\to\mathsf{Y}$
which commutes with the definable morphisms to $h_{S}$,

$[\mathsf{X}\to h_{S},\sigma]=[\mathsf{Y}\to
h_{S},\sigma|_{\mathsf{Y}}]+[\mathsf{X}\setminus\mathsf{Y}\to
h_{S},\sigma|_{\mathsf{X}\setminus\mathsf{Y}}]$

for $\mathsf{Y}$ being $\sigma$-stable definable subassignment of
$\mathsf{X}$, and

$[\mathsf{X}\times h_{\mathbb{A}_{k}^{m}}\to h_{S},\sigma]=[\mathsf{X}\times
h_{\mathbb{A}_{k}^{m}}\to h_{S},\sigma^{\prime}]$

if $\sigma$ and $\sigma^{\prime}$ lift the same $G$-action on $\mathsf{X}$ to
an affine action on $\mathsf{X}\times h_{\mathbb{A}_{k}^{m}}$, for any $m\geq
0$. As above, with respect to fiber product of subassignments endowed with
diagonal $G$-action, the group $K_{0}^{G}(\mathrm{RDef}_{h_{S}})$ is a
commutative rings with unity.

The Grothendieck rings
$K_{0}(\mathrm{RDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
and
$K_{0}(\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
of the categories
$\mathrm{RDef}_{\mathsf{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$
and
$\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})$,
respectively, are defined in the same way as previous.

######  Lemma 4.4.

For any $k$-variety $S$, there are canonical isomorphisms

$K_{0}(\mathrm{RDef}_{h_{S}}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\cong
K_{0}(\mathrm{Var}_{S})$

and

$K_{0}(\mathrm{RDef}_{h_{S}}^{G}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\cong
K_{0}^{G}(\mathrm{Var}_{S}).$

###### Proof.

This statement is a direct corollary of Theorem 4.2 and Lemma 4.3. We can also
refer to [4, Section 16.2] for a proof of the first isomorphism. ∎

##  5\. Integrable functions and measurable subassignments

###  5.1. Rings of motivic functions and Functions

Let $\mathsf{S}$ be a definable subassignment. Put

$\mathbb{A}:=\mathbb{Z}\left[\mathbb{L},\mathbb{L}^{-1},\left(\frac{1}{1-\mathbb{L}^{-n}}\right)_{n\in\mathbb{N}^{*}}\right],$

where by abuse of notation $\mathbb{L}$ also stands for the class of
$\mathsf{S}\times h_{\mathbb{A}_{k}^{1}}$ in
$K_{0}(\mathrm{RDef}_{\mathsf{S}})$. By [4], for any real number $q>1$, there
is a unique morphism of rings

$v_{q}:\mathbb{A}\to\mathbb{R}$

sending $\mathbb{L}$ to $q$, and such that, whenever $q$ is transcendental,
$v_{q}$ is injective. Denote by $\mathbb{A}_{+}$ the subset of $\mathbb{A}$
consisting of elements $a$ with $v_{q}(a)\geq 0$.

We now recall Section 4.6 of [4]. Denote by $|\mathsf{S}|$ the set of points
of $\mathsf{S}$. Let $\mathscr{P}(\mathsf{S})$ be the subring of the ring of
functions $|\mathsf{S}|\to\mathbb{A}$ which is generated by constant functions

$|\mathsf{S}|\to\mathbb{A},$

by functions

$\tilde{\alpha}:|\mathsf{S}|\to\mathbb{Z},$

and by functions

$\mathbb{L}^{\tilde{\beta}}:|\mathsf{S}|\to\mathbb{A},$

for definable morphisms $\alpha,\beta:\mathsf{S}\to h_{\mathbb{Z}}=h[0,0,1]$.
Here, notice that to any definable morphism

$\alpha:\mathsf{S}\to h[0,0,1]$

corresponds a function

$\tilde{\alpha}:|\mathsf{S}|\to\mathbb{Z}.$

Denote by $\mathscr{P}_{+}(\mathsf{S})$ the semiring of functions in
$\mathscr{P}(\mathsf{S})$ with values in $\mathbb{A}_{+}$. In particular, the
ring $\mathscr{P}(h_{\mathrm{Spec}k})$ and the semiring
$\mathscr{P}_{+}(h_{\mathrm{Spec}k})$ are nothing but $\mathbb{A}$ and
$\mathbb{A}_{+}$, respectively. Denote by $\mathscr{P}^{0}(\mathsf{S})$ the
subring of $\mathscr{P}(\mathsf{S})$ which is generated by $\mathbb{L}-1$ and
by character functions $\mathbf{1}_{\mathsf{X}}$ for all definable
subassignments $\mathsf{X}$ of $\mathsf{S}$, and also define

$\mathscr{P}_{+}^{0}(\mathsf{S}):=\mathscr{P}^{0}(\mathsf{S})\cap\mathscr{P}_{+}(\mathsf{S}).$

According to [4, Section 5.3], the semiring $\mathscr{C}_{+}(\mathsf{S})$ of
positive constructible motivic functions on $\mathsf{S}$ and the ring
$\mathscr{C}(\mathsf{S})$ of constructible motivic functions on $\mathsf{S}$
are defined as follows

(5.1) $\displaystyle\mathscr{C}_{+}(\mathsf{S})$
$\displaystyle:=SK_{0}(\mathrm{RDef}_{\mathsf{S}})\otimes_{\mathscr{P}_{+}^{0}(\mathsf{S})}\mathscr{P}_{+}(\mathsf{S}),$
$\displaystyle\mathscr{C}(\mathsf{S})$
$\displaystyle:=K_{0}(\mathrm{RDef}_{\mathsf{S}})\otimes_{\mathscr{P}^{0}(\mathsf{S})}\mathscr{P}(\mathsf{S}).$

If $S$ is an algebraic $k$-variety endowed with a good $\hat{\mu}$-action, we
define

$\mathscr{C}^{\hat{\mu}}(h_{S}):=K_{0}^{\hat{\mu}}(\mathrm{RDef}_{h_{S}})\otimes_{\mathscr{P}^{0}(h_{S})}\mathscr{P}(h_{S}).$

As mentioned in Section 16.1 of [4], with respect to the language
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$ and theory $T_{\mathrm{acl}}$ we can
define rings
$\mathscr{P}_{+}(\mathsf{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$,
$\mathscr{P}(\mathsf{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$,
$\mathscr{C}_{+}(\mathsf{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$,
and in the same way, the ring
$\mathscr{C}^{\hat{\mu}}(h_{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$.

In the rest of the article we will not work with the rings
$\mathscr{C}(h_{S})$,
$\mathscr{C}(h_{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
and
$\mathscr{C}^{\hat{\mu}}(h_{S},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
except the trivial case $S=\mathrm{Spec}k$. For this trivial case, we have the
following lemma, which is is obvious from the definition.

######  Lemma 5.1.

There exist canonical isomorphisms

$\displaystyle\mathscr{C}_{+}(h_{\mathrm{Spec}k})$ $\displaystyle\cong
SK_{0}(\mathrm{RDef}_{k})\otimes_{\mathbb{N}[\mathbb{L}-1]}\mathbb{A}_{+},$
$\displaystyle\mathscr{C}(h_{\mathrm{Spec}k},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
$\displaystyle\cong\mathscr{M}_{\mathrm{loc}},$
$\displaystyle\mathscr{C}^{\hat{\mu}}(h_{\mathrm{Spec}k},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
$\displaystyle\cong\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}.$

The important properties of $\mathscr{C}_{+}(\mathsf{S})$ and
$\mathscr{C}(\mathsf{S})$ are given in Section 5 of [4].

According to [4, Section 3], the K-dimension of a definable subassignment
(with K a capital letter not a mathematical notation) is defined as follows.
If $\mathsf{S}$ is a definable subassignment of $h_{\mathcal{X}}$, with
$\mathcal{X}$ being an algebraic $k(\\!(t)\\!)$-variety, then the K-dimension
of $\mathsf{S}$, denoted by $\text{Kdim}\mathsf{S}$, is the dimension of the
$k(\\!(t)\\!)$-variety which is the intersection of all
$k(\\!(t)\\!)$-subvarieties $\mathcal{Y}$ of $\mathcal{X}$ with
$h_{\mathcal{Y}}$ containing $\mathsf{S}$. It may happen that the intersection
is empty; in that case, we define

$\text{Kdim}\mathsf{S}:=-\infty.$

If $\mathsf{S}$ is a definable subassignment of $h_{\mathcal{X}\times
X\times\mathbb{Z}^{r}}$, with $\mathcal{X}$ as above and $X$ being a
$k$-variety, then we define

$\text{Kdim}\mathsf{S}:=\text{Kdim}\mathsf{pr}_{1}(\mathsf{S}),$

where

$\mathsf{pr}_{1}:h_{\mathcal{X}\times X\times\mathbb{Z}^{r}}\to
h_{\mathcal{X}}$

is the first projection.

A positive constructible motivic function $\varphi$ in
$\mathscr{C}_{+}(\mathsf{S})$ is called of K-dimension $\leq d$ if $\varphi$
is a finite sum $\sum_{i}\alpha_{i}\mathbf{1}_{\mathsf{S}_{i}}$ in
$\mathscr{C}_{+}(\mathsf{S})$ such that the K-dimension of every
$\mathsf{S}_{i}$ is $\leq d$. Let $\mathscr{C}_{+}^{\leq d}(\mathsf{S})$ be
the sub-semigroup of $\mathscr{C}_{+}(\mathsf{S})$ of elements of K-dimension
$\leq d$, and

$C_{+}^{d}(\mathsf{S}):=\mathscr{C}_{+}^{\leq
d}(\mathsf{S})/\mathscr{C}_{+}^{\leq d-1}(\mathsf{S})$

and

$C_{+}(\mathsf{S}):=\bigoplus_{d\geq 0}C_{+}^{d}(\mathsf{S}).$

An element in $C_{+}(\mathsf{S})$ is called a positive constructible motivic
Function on $\mathsf{S}$ (with the capital letter F). Clearly,
$C_{+}(\mathsf{S})$ is a graded abelian semigroup and has a module structure
over the semiring $\mathscr{C}_{+}(\mathsf{S})$ (cf. [4, Section 6]).

###  5.2. Integrable positive Functions and measurable subassignments

Let $\mathsf{S}$ be in $\mathrm{Def}_{k}$. By [4, Theorem 10.1.1], there
exists a unique functor $\mathrm{I}_{\mathsf{S}}C_{+}$ from the category
$\mathrm{Def}_{\mathsf{S}}$ to the category of abelian semigroups which sends
every morphism

$\mathsf{f}:\mathsf{X}\to\mathsf{Y}$

in $\mathrm{Def}_{\mathsf{S}}$ to a morphism of semigroups

$\mathsf{f}_{!}:\mathrm{I}_{\mathsf{S}}C_{+}(\mathsf{X})\to\mathrm{I}_{\mathsf{S}}C_{+}(\mathsf{Y})$

and satisfies the axioms A0–A8. If $\mathsf{S}=h_{\mathrm{Spec}k}$ we write
$\mathrm{I}C_{+}(\mathsf{X})$ instead of
$\mathrm{I}_{\mathsf{S}}C_{+}(\mathsf{X})$, and call it the semigroup of
integrable positive Functions on $\mathsf{X}$. By Proposition 12.2.2 of [4],
if $\mathsf{X}$ is a definable subassignment of $h[m,n,0]$ which is bounded,
i.e., there exists an $s\in\mathbb{N}$ such that $\mathsf{X}$ is contained in
the subassignment of $h[m,n,0]$ defined by

$\mathrm{ord}_{t}x_{i}\geq-s$

for all $1\leq i\leq m$, then $[\mathbf{1}_{\mathsf{X}}]$ belongs to
$\mathrm{I}C_{+}(\mathsf{X})$, where $\mathbf{1}_{\mathsf{X}}$ is the
characteristic function on $\mathsf{X}$. (In the previous definition of
boundedness, if $s=0$ then $\mathsf{X}$ is said to be positively bounded.)

Also in the trivial case $\mathsf{S}=h_{\mathrm{Spec}k}$, let us take
$\mathsf{f}$ to be the projection of $\mathsf{X}$ onto the final subassignment
$h_{\mathrm{Spec}k}$ of $\mathrm{Def}_{k}$. We denote by $\widetilde{\mu}$ the
morphism of semigroups $\mathsf{f}_{!}$, namely,

$\widetilde{\mu}:\mathrm{I}C_{+}(\mathsf{X})\to\mathrm{I}C_{+}(h_{\mathrm{Spec}k})\cong\mathscr{C}_{+}(h_{\mathrm{Spec}k}).$

Applying Section 16.1 of [4] we have a canonical morphism of rings

$\mathscr{C}_{+}(h_{\mathrm{Spec}k})\to\mathscr{C}_{+}(h_{\mathrm{Spec}k},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}})).$

On the other hand, we also have another canonical morphism

$\mathscr{C}_{+}(h_{\mathrm{Spec}k},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\to\mathscr{C}(h_{\mathrm{Spec}k},(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))\cong\mathscr{M}_{\mathrm{loc}}.$

Taking the composition of the last two morphisms with $\widetilde{\mu}$ we get
a morphism of rings

$\mu:\mathrm{I}C_{+}(\mathsf{X})\to\mathscr{M}_{\mathrm{loc}}.$

As mentioned previously, if $\mathsf{X}$ is a bounded definable subassignment
in $\mathrm{Def}_{k}$, then $[\mathbf{1}_{\mathsf{X}}]$ is in
$\mathrm{I}C_{+}(\mathsf{X})$. In this case, we call $\mathsf{X}$ a
motivically measurable (definable) subaasignment. We define the motivic
measure of $\mathsf{X}$ to be

$\mu(\mathsf{X}):=\mu([\mathbf{1}_{\mathsf{X}}]),$

which lies in $\mathscr{M}_{\mathrm{loc}}$. By the additivity of the integral
(Axiom A2 in [4, Theorem 10.1.1]), the motivic measure $\mu$ is additive on
bounded definable subassignments.

Denote by $\widehat{\mathscr{M}}_{k}$ the completion of $\mathscr{M}_{k}$ in
the sense of [8], and by $\delta$ the canonical morphism
$\mathscr{M}_{\mathrm{loc}}\to\widehat{\mathscr{M}}_{k}$ defined by the
expansion of $1-\mathbb{L}^{-n}$, for every $n$ in $\mathbb{N}^{*}$.

######  Proposition 5.2.

Let $X$ be an algebraic $k$-variety, $A$ a semi-algebraic subset of
$\mathscr{L}(X)$, and $\mathsf{A}$ the small definable subassignment
corresponding to $A$ via the equivalence of categories between
$\mathrm{SA}_{k}(X)$ and
$\mathrm{SDef}_{k}(X,(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
in Theorem 4.1. Then

$\delta(\mu(\mathsf{A}))=\mu^{\prime}(A),$

where $\mu^{\prime}$ is Denef-Loeser’s motivic volume defined in [8].

###### Proof.

Note that if a definable function

$\alpha:\mathsf{A}\to h_{\mathbb{Z}}$

in the language $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k)$ is the zero function
on $\mathsf{A}$, then the semi-algebraic function

$\tilde{\alpha}:A\to\mathbb{Z}$

corresponding to $\alpha$ via the equivalence of categories of
$\mathrm{SA}_{k}(X)$ and
$\mathrm{SDef}_{k}(X,(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))$
in Theorem 4.1 is also the zero function on $A$. Now applying Theorem 16.3.1
of [4] to $\alpha=0$ we get the proposition. ∎

###  5.3. Invariant definable subassignments and their measure

Let $m,n$ be in $\mathbb{N}$, and $\gamma=(\gamma_{1},\dots,\gamma_{m})$ in
$\mathbb{Z}^{m}$. A definable subassignment $\mathsf{X}$ of $h[m,n,0]$ is
called $\gamma$-invariant if, for every $K$ in $\mathrm{Field}_{k}$, $(a,b)$
and $(x,y)$ in

$h[m,n,0](K)=K(\\!(t)\\!)^{m}\times K^{n}$

satisfying

$\mathrm{ord}_{t}x_{i}\geq\gamma_{i}$

for $1\leq i\leq m$, both elements $(a,b)$ and $(a,b)+(x,y)$ are
simultaneously in either $\mathsf{X}(K)$ or in the complement of
$\mathsf{X}(K)$ in $K(\\!(t)\\!)^{m}\times K^{n}$. A definable subassignment
of $h[m,n,0]$ is called invariant if it is $\gamma$-invariant for some
$\gamma$ in $\mathbb{Z}^{m}$. In the case $\gamma_{i}=\beta\in\mathbb{Z}$ for
all $1\leq i\leq m$, we write $\beta$-invariant instead of $\gamma$-invariant.
Note that if $\mathsf{X}$ is $\gamma$-invariant and
$\gamma_{i}\leq\gamma^{\prime}_{i}$ for all $1\leq i\leq m$, then $\mathsf{X}$
is also $(\gamma^{\prime}_{1},\dots,\gamma^{\prime}_{m})$-invariant. It is a
fact that any bounded definable subassignment of $h[m,0,0]$ closed in the
valuation topology is $\gamma$-invariant for some $\gamma$ in
$\mathbb{Z}^{m}$.

Now, let $\beta$ be in $\mathbb{N}^{*}$, and let $\mathsf{X}$ be a bounded
definable subassignment of $h[m,n,0]$ with $\mathrm{ord}_{t}x_{i}\geq 0$ for
every $x=(x_{1},\dots,x_{m})$ on $\mathsf{X}$ and for all $1\leq i\leq m$.
Then $\mathsf{X}$ is $\beta$-invariant if and only if there exists a
constructible subset $X_{\beta}$ of

$\mathbb{A}_{k}^{\beta
m}\times_{k}\mathbb{A}_{k}^{n}\cong\mathscr{L}_{\beta-1}(\mathbb{A}_{k}^{m})\times\mathbb{A}_{k}^{n}$

such that, for every $K$ in $\mathrm{Field}_{k}$,

$\mathsf{X}(K)=\left(\pi_{\beta}(K)\right)^{-1}\left(X_{\beta}(K)\right),$

which is the pullback of $X_{\beta}(K)$ under the canonical map

$\pi_{\beta}(K):K[[t]]^{m}\times
K^{n}\to\big{(}K[t]/(t^{\beta})\big{)}^{m}\times K^{n}\cong K^{\beta m}\times
K^{n}.$

Indeed, we can show this by observing that, for $K$ in $\mathrm{Field}_{k}$,
every fiber of the restriction $\pi_{\beta}(K)$ is definably bijective to the
set $(t^{\beta})K[[t]]^{m}$. The maps $\pi_{\beta}(K)$ induce the canonical
morphism

$\pi_{\beta}:h[m,n,0]\to h[0,\beta m+n,0],$

and $\mathsf{X}$ is $\beta$-invariant if and only if there exists a
constructible subset $X_{\beta}\subseteq\mathbb{A}_{k}^{\beta
m}\times_{k}\mathbb{A}_{k}^{n}$ such that
$\mathsf{X}=\pi_{\beta}^{-1}(h_{X_{\beta}})$. By abuse of notation, we shall
denote the canonical morphism $\mathsf{X}\to h_{X_{\beta}}$ by $\pi_{\beta}$.

######  Lemma 5.3.

Let $\beta\leq\beta^{\prime}$ be in $\mathbb{N}^{*}$, let $\mathsf{X}$ and
$X_{\beta}$ be as previous. Then the identity

$[h_{X_{\beta^{\prime}}}]=[h_{X_{\beta}}]\mathbb{L}^{(\beta^{\prime}-\beta)m}$

holds in $K_{0}(\mathrm{RDef}_{k})$. As a consequence, the element
$[h_{X_{\beta}}]\mathbb{L}^{-(\beta+1)m}$ in
$K_{0}(\mathrm{RDef}_{k})[\mathbb{L}^{-1}]$ is independent of the choice of
sufficiently large $\beta$.

###### Proof.

The natural map

$\mathscr{L}_{\beta^{\prime}-1}(\mathbb{A}_{k}^{m})\times_{k}\mathbb{A}_{k}^{n}\to\mathscr{L}_{\beta-1}(\mathbb{A}_{k}^{m})\times_{k}\mathbb{A}_{k}^{n}$

induced by truncation is a Zariski locally trivial fibration with fiber
$\mathbb{A}_{k}^{(\beta^{\prime}-\beta)m}$. Along this map,
$X_{\beta^{\prime}}$ is the preimage of $X_{\beta}$. Thus we get the identity
$[h_{X_{\beta^{\prime}}}]=[h_{X_{\beta}}]\mathbb{L}^{(\beta^{\prime}-\beta)m}$
in $K_{0}(\mathrm{RDef}_{k})$. ∎

We denote by $\mathrm{vol}(\mathsf{X})$ the image of
$[h_{X_{\beta}}]\mathbb{L}^{-(\beta+1)m}\in
K_{0}(\mathrm{RDef}_{k})[\mathbb{L}^{-1}]$ under the canonical morphism (due
to [4, Section 16.1])

$K_{0}(\mathrm{RDef}_{k})[\mathbb{L}^{-1}]\to
K_{0}(\mathrm{RDef}_{k}(\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k),T_{\mathrm{acl}}))[\mathbb{L}^{-1}]\cong\mathscr{M}_{k}.$

######  Theorem 5.4.

Let $\mathsf{X}$ be an invariant bounded definable subassignment of $h[m,n,0]$
such that, for every $(x,y)$ on $\mathsf{X}$ with $x=(x_{1},\dots,x_{m})$,
$\mathrm{ord}_{t}x_{i}\geq 0$ for all $1\leq i\leq m$. Then the identity

$\mu(\mathsf{X})=\mathrm{loc}(\mathrm{vol}(\mathsf{X}))$

holds in $\mathscr{M}_{\mathrm{loc}}$.

###### Proof.

Assume that $\mathsf{X}$ is $\beta$-invariant, for some $\beta$ in
$\mathbb{N}^{*}$. For $k(\\!(t)\\!)$-coordinates $(x_{1},\dots,x_{m})$ in
$\mathsf{X}$, let us write

$x_{i}=a_{i0}+a_{i1}t+\cdots+a_{i,\beta-1}t^{\beta-1}+\cdots,\quad 1\leq i\leq
m.$

Consider the inclusion

$\mathsf{i}:h[m,n,0]\hookrightarrow h[m,\beta m+n,0]$

given by $(x,y)\mapsto(x,(a_{i0},a_{i1},\dots,a_{i,\beta-1})_{1\leq i\leq
m},y)$, and the projection

$\mathsf{pr}:h[m,\beta m+n,0]\to h[0,\beta m+n,0]$

given by $(x,z)\mapsto z$. Denote by $\mathsf{i}_{\mathsf{X}}$ the restriction
of $\mathsf{i}$ on $\mathsf{X}$. We can regard $\mathsf{i}_{\mathsf{X}}$ as an
inclusion

$\mathsf{i}_{\mathsf{X}}:\mathsf{X}\to\mathsf{X}[0,\beta m,0].$

Denote by $\mathsf{pr}_{\mathsf{X}}$ the restriction of $\mathsf{pr}$ on
$\mathsf{X}[0,\beta m,0]$. Then the composition
$\mathsf{pr}_{\mathsf{X}}\circ\mathsf{i}_{\mathsf{X}}$ is nothing but the
canonical map $\pi_{\beta}:\mathsf{X}\to h_{X_{\beta}}$. By the functoriality
(Axiom A0) of the integral in [4, Theorem 10.1.1] we have
$(\pi_{\beta})_{!}=(\mathsf{pr}_{\mathsf{X}})_{!}\circ(\mathsf{i}_{\mathsf{X}})_{!}$,
hence

$(\pi_{\beta})_{!}\big{(}\big{[}\mathbf{1}_{\mathsf{X}}\big{]}\big{)}=(\mathsf{pr}_{\mathsf{X}})_{!}\big{(}\big{[}\mathbf{1}_{\mathsf{i}(\mathsf{X})}\big{]}\big{)}.$

Since $\mathsf{X}$ is $\beta$-invariant, by fixing an element $(a,b)$ in
$\mathsf{i}(\mathsf{X})$ we have

$\mathsf{i}(\mathsf{X})=\mathsf{pr}_{\mathsf{X}}^{-1}(h_{X_{\beta}})=\left\\{(a,b)+(x,y)\in
h[m,\beta m+n,0]\mid\mathrm{ord}_{t}x_{i}\geq\beta\ \forall\ 1\leq i\leq
m\right\\}.$

By definition, constructible motivic Functions on $\mathsf{i}(\mathsf{X})$ are
equivalence classes of elements of $C_{+}(\mathsf{i}(\mathsf{X}))$ modulo
support of smaller dimension (cf. [3, Section 3.3], [4, Section 6]), hence in
$\mathrm{I}C_{+}(\mathsf{i}(\mathsf{X}))$ we have
$\big{[}\mathbf{1}_{\mathsf{i}(\mathsf{X})}\big{]}=\big{[}\mathbf{1}_{\widetilde{\mathsf{X}}}\big{]}$,
where $\widetilde{\mathsf{X}}$ is defined similarly as
$\mathsf{i}(\mathsf{X})$ with $\mathrm{ord}_{t}x_{i}=\beta$ replacing
$\mathrm{ord}_{t}x_{i}\geq\beta$. Now, applying Axiom A7 in [4, Theorem
10.1.1] inductively, we get

$(\mathsf{pr}_{\mathsf{X}})_{!}\big{(}\big{[}\mathbf{1}_{\mathsf{i}(\mathsf{X})}\big{]}\big{)}=\mathbb{L}^{-(\beta+1)m}\big{[}\mathbf{1}_{h_{X_{\beta}}}\big{]}.$

The projection $\mathsf{f}$ of $\mathsf{X}$ onto the final object
$h_{\mathrm{Spec}k}$ in $\mathrm{Def}_{k}$ can be factorized through the
canonical map $\pi_{\beta}:\mathsf{X}\to h_{X_{\beta}}$, namely, we have the
following commutative diagram

$\textstyle{\mathsf{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\beta}}$$\scriptstyle{\mathsf{f}}$$\textstyle{h_{X_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\mathsf{f}}}$$\textstyle{h_{\mathrm{Spec}k}.}$

Therefore,

$\mu(\mathsf{X})=\mathsf{f}_{!}([\mathbf{1}_{\mathsf{X}}])=\widetilde{\mathsf{f}}\big{(}\mathbb{L}^{-(\beta+1)m}\big{[}\mathbf{1}_{h_{X_{\beta}}}\big{]}\big{)}.$

By [4, Proposition 5.3.1], we have

$\displaystyle\mathscr{C}_{+}(h[0,\beta m,0])$ $\displaystyle\cong
SK_{0}(\mathrm{RDef}_{h[0,\beta
m,0]})\otimes_{\mathscr{P}_{+}^{0}(h_{\mathrm{Spec}k})}\mathscr{P}_{+}(h_{\mathrm{Spec}k})$
$\displaystyle\cong SK_{0}(\mathrm{RDef}_{h[0,\beta
m,0]})\otimes_{\mathbb{N}[\mathbb{L}-1]}\mathbb{A}_{+}.$

Thus, the element $\big{[}\mathbf{1}_{h_{X_{\beta}}}\big{]}$ in
$\mathscr{C}_{+}(h_{X_{\beta}})$ can be written as $\big{[}h_{X_{\beta}}\to
h_{X_{\beta}}\big{]}\otimes 1$. Then we have the identities

$\mu(\mathsf{X})=\mathbb{L}^{-(\beta+1)m}\big{[}h_{X_{\beta}}\to
h_{X_{\beta}}\to
h_{\mathrm{Spec}k}\big{]}=\mathbb{L}^{-(\beta+1)m}\big{[}h_{X_{\beta}}\big{]}=\mathrm{loc}(\mathrm{vol}(\mathsf{X}))$

hold true in $\mathscr{M}_{\mathrm{loc}}$. ∎

Let us recall some settings in Section 14.5 of [4] and Section 4.3 of [13] on
the ramification. Consider a formula $\varphi$ in the language
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$, i.e., the coefficients of
$\varphi$ are in $k[t]$ in the valued field sort, in $k$ in the residue field
sort, such that $\varphi$ has $m$ free variables in the valued field sort, $n$
free variables in residue field sort, and $r$ free variables in the value
group sort. For each $e$ in $\mathbb{N}^{*}$, let $\varphi[e]$ denote the
formula obtained from $\varphi$ by replacing everywhere $t$ by $t^{e}$. If
$\mathsf{X}$ is a definable subassignment of $h[m,n,r]$ defined by $\varphi$,
we denote by $\mathsf{X}[e]$ the definable subassignment of $h[m,n,r]$ defined
by $\varphi[e]$. Thus, if in addition $\mathsf{X}$ is bounded, then so is
$\mathsf{X}[e]$, that is, $[\mathbf{1}_{\mathsf{X}[e]}]$ is in
$\mathrm{I}C_{+}(\mathsf{X}[e])$, for every $e$ in $\mathbb{N}^{*}$ (cf. [4,
Proposition 14.5.1]).

######  Proposition 5.5 ([4], Theorem 14.5.3).

Assume $\mathsf{X}$ is a bounded definable subassignment of $h[m,0,0]$ defined
by a formula in $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$ with $m$ free
variables in the valued field sort. Then the formal power series

$Z_{\mathsf{X}}(T)=\sum_{e\in\mathbb{N}^{*}}\mu(\mathsf{X}[e])T^{e}$

is in $\mathscr{M}_{\mathrm{loc}}[[T]]_{\mathrm{sr}}$.

Let $\beta$ be in $\mathbb{N}^{*}$, and $\mathsf{X}$ a $\beta$-invariant
bounded definable subassignment of $h[m,n,0]$ defined by a formula in
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$. Then, for every $e$ in
$\mathbb{N}^{*}$, the bounded definable subassignment $\mathsf{X}[e]$ of
$h[m,n,0]$ is $\beta e$-invariant; therefore, there exists a constructible
subset $X_{\beta e}$ of $\mathscr{L}_{\beta
e-1}(\mathbb{A}_{k}^{m})\times_{k}\mathbb{A}_{k}^{n}$ such that, for every $K$
in $\mathrm{Field}_{k}$, $\mathsf{X}[e](K)$ is the pullback of $X_{\beta
e}(K)$ under the canonical map

$K(\\!(t)\\!)^{m}\times K^{n}\to\big{(}K[t]/(t^{\beta e})\big{)}^{m}\times
K^{n}.$

######  Proposition 5.6.

Let $\beta$ be in $\mathbb{N}^{*}$, and let $\mathsf{X}$ be a
$\beta$-invariant bounded definable subassignment of $h[m,n,0]$ defined by a
formula $\varphi$ in $\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$ not
containing the symbol $\overline{\mathrm{ac}}$. Then the definable
subassignment $h_{X_{\beta e}}$ is stable by the natural $\mu_{e}$-action on
$h[m,n,0]$ defined by

$\lambda\cdot(x,\xi)=(\lambda x,\xi),\quad\text{with}\quad\lambda
x(t)=x(\lambda t).$

As a consequence, the quantity $\mathrm{vol}({\mathsf{X}}[e])$ is an element
in $\mathscr{M}_{k}^{\mu_{e}}$, and the identity

$\mu(\mathsf{X}[e])=\mathrm{loc}(\mathrm{vol}({\mathsf{X}}[e]))$

holds in $\mathscr{M}_{\mathrm{loc}}^{\mu_{e}}$, thus the series

$Z_{\mathsf{X}}(T)=\sum_{e\in\mathbb{N}^{*}}\mu(\mathsf{X}[e])T^{e}$

is in $\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}[[T]]_{\mathrm{sr}}$.

###### Proof.

By definition of $\beta$-invariance, we have

$h_{X_{\beta e}}\cong\left\\{(x,\xi)\in
h[m,n,0]\mid\varphi[e](x,\xi),0\leq\mathrm{ord}_{t}x_{i}\leq\beta e-1,1\leq
i\leq m\right\\}.$

By Corollary 2.1.2 of [4], the
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$-formula $\varphi$ is a finite
disjunction of formulas of the form

$\psi(\overline{\mathrm{ac}}g^{\prime}_{1}(x),\dots,\overline{\mathrm{ac}}g^{\prime}_{l^{\prime}}(x),\xi)\wedge\vartheta(\mathrm{ord}_{t}f^{\prime}_{1}(x),\dots,\mathrm{ord}_{t}f^{\prime}_{l}(x),\alpha),$

where $f^{\prime}_{i}$ and $g^{\prime}_{j}$ are polynomials over $k[t]$,
$\psi$ is an $\mathbf{L}_{\mathrm{Rings}}$-formula, and $\vartheta$ is an
$\mathbf{L}_{\mathrm{PR}}$-formula. Since $\varphi$ is equivalent to an
$\mathcal{L}_{\mathrm{DP},\mathrm{P}}(k[t])$-formula without angular component
symbol $\overline{\mathrm{ac}}$ due to the hypothesis, $\varphi[e]$ is a
finite disjunction of formulas of the form

$\psi_{1}((g_{j}(x))_{j})\wedge\psi_{2}(\xi)\wedge\vartheta((\mathrm{ord}_{t}f_{i}(x))_{i},\alpha),$

where $f_{i}$ and $g_{j}$ are polynomials over $k[t^{e}]$, $\psi_{1}$ and
$\psi_{2}$ are $\mathbf{L}_{\mathrm{Rings}}$-formulas, and $\vartheta$ is an
$\mathbf{L}_{\mathrm{PR}}$-formula. If in $f_{i}(x(t))$ and $g_{j}(x(t))$ we
replace $t$ by $\lambda t$, for any $x(t)$ in $K(\\!(t)\\!)^{m}$, $\lambda$ in
$\mu_{e}(K)$, $K$ in $\mathrm{Field}_{k}$, then we get expressions of
$f_{i}(x(\lambda t))$ and $g_{j}(x(\lambda t))$, since the coefficients of
polynomials $f_{i}$ and $g_{j}$ in $k[t^{e}][x]$ do not change. This proves
that if $(x,\xi)$ is in $h_{X_{\beta e}}$, so is $(\lambda x,\xi)$, for any
$\lambda$ in $\mu_{e}$, that is, $h_{X_{\beta e}}$ is stable under the action
of $\mu_{e}$. ∎

Now assume that $\mathsf{X}$ is small. As mentioned above, there is a
canonical action of $\hat{\mu}$ on $h[m,0,0]$ induced by

$(\lambda,t)\mapsto\lambda t.$

We say that the definable subassignment $\mathsf{X}$ is $\hat{\mu}$-stable if
there exists an $n\in\mathbb{N}^{*}$ such that, for every
$x=(x_{1}(t),\dots,x_{m}(t))$ in $\mathsf{X}$ and $\lambda$ in $\mu_{n}$, the
point

$\lambda\cdot x=(x_{1}(\lambda t),\dots,x_{m}(\lambda t))$

is in $\mathsf{X}$. Since formulas defining $\mathsf{X}$ is in the Denef-Pas
language, by quantifier elimination for algebraically closed fields, they also
define a semi-algebraic subset $X$ of some arc space
$\mathscr{L}(\mathbb{A}_{k}^{m})$ of $\mathbb{A}_{k}^{m}$. The assignment

$\mathsf{X}\mapsto X$

carries the canonical $\hat{\mu}$-action on $h[m,0,0]$ to the canonical
$\hat{\mu}$-action on $\mathscr{L}(\mathbb{A}_{k}^{m})$, and in that way, $X$
is also $\hat{\mu}$-stable in $\mathscr{L}(\mathbb{A}_{k}^{m})$. As in [4,
Theorem 16.3.1, Remark 16.3.2] we can see that $X$ is measurable as
$\mathsf{X}$ is measurable, and that since with the above action
$\mu^{\prime}(X)$ is in $\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}$ the measure
$\mu(\mathsf{X})$ of $\mathsf{X}$ is also in
$\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}$. Here, as in [4, Theorem 16.3.1],
$\mu^{\prime}$ stands for Denef-Loeser’s motivic measure [8], and further by
[4, Remark 16.3.2], we can consider that this measure takes value in
$\mathscr{M}_{\mathrm{loc}}^{\hat{\mu}}$ for the context with
$\hat{\mu}$-action.

######  Acknowledgement.

Part of the present paper was worked out at the Vietnam Institute for Advanced
Study in Mathematics (VIASM), the author would like to thank the institute for
warm hospitality.

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