Motivic integrals and functional equations

Abstract

Abstract. A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived by using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant, and there is a simple recursive algorithm to find its coefficients. The method is fairly universal and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered collections of arcs. §1. Introduction Motivic integration, introduced by M. Kontsevich, is a powerful tool for exploring the space of formal arcs on a given variety. Motivic integrals provide the generating series for motivic measures of level sets of some arc invariants. There are examples of such integrals that can be calculated explicitly (see, e.g., [1]). In more general situations the values of such integrals are unknown; however, some of them satisfy functional equations if some auxiliary variables are introduced.In this paper, with the help of the Denef-Loeser formula for the change of variables (see [1]), a functional equation is derived for the motivic integral that gives the generating series corresponding to the Milnor number of a plane curve. Its solution is unique up to multiplication by a constant, and there is a simple algorithm to express the solution's coefficients via the initial data. For example, this means that the solution satisfies some partial differential equations. The equation obtained gives a method for computing the motivic measure (and consequently, the Hodge-Deligne polynomial) of the stratum {µ = const} in the space of plane curves. Some examples are considered in §4.A similar idea gives some other equations, for example, an equation for the integral corresponding to the intersection number over the space of pairs of arcs. Moreover, using the notion of the power structure on the Grothendieck ring (see [2]), we introduce a motivic measure on the space of unordered collections of arcs. We also derive a curious equation for the integral corresponding to the intersection number in this case.Some generating series with coefficients in the Grothendieck ring of varieties K 0 (V ar C ) (or in the Grothendieck ring of Chow motives) satisfy functional equations similar to the functional equation for the Hasse-Weil zeta function. These equations, obtained by M. Kapranov [3] and F. Heinloth [4], follow from the duality theory on curves and Abelian varieties. It should be noted that they have an origin different from our approach. §2. Motivic measure Let L = L C 2 ,0 be the space of arcs at the origin on the plane. It is the set of pairs (x(t), y(t)) of formal power series (without degree 0 terms).2000 Mathematics Subject Classification. Primary 32S45, 28B10.

Symmetries of some motivic integrals

“…If orders of the series x(t) and y(t) are equal, then y(t) = λx(t) + y ′ (t), where λ is a nonzero number , and the order of y ′ (t) is greater than the one of y(t). This implies the proposition (5). Now…”

“…It is shown in [5], that its solution can be completely reconstructed from the initial condition (2) .…”

Симметрии Мотивных Интегралов

“…Симметрии мотивных интегралов Е. А. Горский Пусть µ(γ) -число Милнора [1] ростка плоской кривой γ(t) = (x(t), y(t)). Рассмотрим семейство мотивных интегралов [2] G k,m (t) = {Ord x(t)=k, Ord y(t)=m} t µ(γ) dγ по пространству параметризованных дуг на плоскости (C 2 , 0).…”

On the Motivic Measure on the Space of Functions

“…The author do not know the simple explanation of this fact. For example, the same integral over the space L * /C * itself is much more complicated and satisfy some functional equations ( [6]). One can say that addition of all other symmetric powers "simplify" the integral.…”