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Constructible motivic functions and motivic integration. (English) Zbl 1179.14011

This article presents a new approach to motivic integration. The most important new feature is probably that parametrized integrals can be treated (such that the Fubini theorem holds). Another improvement compared to “classical” motivic integration is that the ring where the integrals take values is constructed without a completion process.
I will now recall what motivic integration is and at the same time roughly explain the approach used in this article. The article makes essential use of model theory, in particular of cell decomposition. For the reader not familiar with this, let me just say: formulas are a generalization of algebraic sets (like constructible sets, but even further), and cell decomposition is a way to decompose such a set into finitely many subsets with some nice properties.
Suppose that \(\phi\) is a formula in some language of valued fields. On each \(p\)-adic field \(\mathbb{Q}_p\), we have the Haar measure \(\mu_p\), so we may consider the measure \(\mu_p(\phi(\mathbb{Q}_p))\). One way to effectively compute this measure is the following. Using cell decomposition, we can reduce computing the measure to computing integrals over fibers of cells, and by definition of a cell, such an integral can easily be seen to be equal to a geometric series.
Now for one fixed formula \(\phi\), we can do this computation for different primes \(p\). (For non model-theorists: think of the algebraic set being defined over \(\mathbb{Z}\).) It turns out that with exception of finitely many primes, the computation which we carry out is always essentially the same. The only reasons for \(\mu_p(\phi(\mathbb{Q}_p))\) to depend on \(p\) are that at some places, we plug \(p\) into our computation, and at some other places, we have to compute the number of solutions of a formula over the residue field \(\mathbb{F}_p\).
Motivic integration (or rather motivic measuring) can be seen as carrying out the above computation without fixing \(p\). Instead of computing a number, we work in a ring \(\mathcal{R}\) containing a symbol \(\mathbb{L}\) and one symbol \([\psi]\) for each ring formula \(\psi\). Each time \(p\) would appear in the computation of the \(p\)-adic measure, we use \(\mathbb{L}\) instead, and each time we have to count the points of a set \(\psi(\mathbb{F}_p)\), we use \([\psi]\). The resulting element of \(\mathcal{R}\) is what we call the motivic measure \(\mu(\phi)\) of our original formula \(\phi\). By construction, this motivic measure “knows” the \(p\)-adic measure of \(\phi(\mathbb{Q}_p)\) for almost all \(p\): simply take the image of \(\mu(\phi)\) under the map \(\mathcal{R} \to \mathbb{Q}\) sending \(\mathbb{L}\) to \(p\) and \([\psi]\) to \(|\psi(\mathbb{F}_p)|\).
Note that although in our computation of \(p\)-adic measure, we had to compute geometric series, no completion process is needed in the definition of \(\mathcal{R}\): these series are so explicit that we only need \(\mathcal{R}\) to contain the inverses of some polynomials in \(\mathbb{L}\) to make sense of the geometric series in this abstract setting.
Apart from yielding the \(p\)-adic measure for almost all \(p\) in a uniform way, this motivic measure has another advantage: it gives sense to measure in a much larger class of valued fields. Indeed, using Denef-Pas cell decomposition, it can be carried out in \(k((t))\), where \(k\) is any field of characteristic \(0\). This is the context of the present article.
One of the difficulties of this definition of motivic integration is to prove that the result is well defined. In particular, it should not depend on the order of the coordinates which one chooses for the cell decomposition. Proving this independence is one of the key points of the article.
Another difficulty is that when computing the measure of a cell, we do this coordinate by coordinate, so in intermediate steps, we need to be able to integrate some kind of \(\mathcal{R}\)-valued functions. However, if we would use ordinary \(\mathcal{R}\)-valued functions, we would loose too much information (see the example given in Section 1.4 of the introduction). Finding a working notion of “\(\mathcal{R}\)-valued function” is one of the big achievements of the article. This definition looks quite complicated and technical. However, the reward for this work is that once it is done, one automatically gets a nice framework which allows to do all the computations of measures and integrals as functions of parameters, and where theorems like Fubini and change of variables hold as one would expect. (The enhanced \(\mathcal{R}\)-valued functions on a set \(S\) are called “constructible functions” in the article and they are denoted by \(\mathcal{C}(S)\); the ring called \(\mathcal{R}\) in this review has no own name, but it can be written as \(\mathcal{C}(\mathrm{point})\).)
Other technical difficulties are: one has to deal with functions which are defined only almost everywhere, and one has to deal with functions which are not integrable (i.e.whose integral would diverge).
Another advantage of this approach is that the set of constructible functions (the functions which can be integrated) can easily be extended. Indeed, in a follow-up article [C. R., Math., Acad. Sci. Paris 341, No. 12, 741–746 (2005; Zbl 1081.14032)] the authors show how additive characters can be added to the framework.

MSC:

14E18 Arcs and motivic integration
14G20 Local ground fields in algebraic geometry
03C10 Quantifier elimination, model completeness, and related topics
11S85 Other nonanalytic theory
14B10 Infinitesimal methods in algebraic geometry

Citations:

Zbl 1081.14032
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References:

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