9. EXTRACTING THE SINGULAR PART 59 Definition 9.2.2. Let PQ((0, 1)) C∞((0, 1)) be the subalgebra of functions of this form. Elements of this subalgebra will be called rational periods. Note that PQ((0, 1)) is, indeed, a subalgebra the sum of functions cor- responds to the disjoint union of the pairs (X, D) of algebraic varieties, and the product of functions corresponds to fibre product of the algebraic va- rieties (X, D) over U. Further, PQ((0, 1)) is of countable dimension over Q. Note that, if f is a rational period, then, for every rational number t Q (0, 1), f(t) is a period in the sense of Kontsevich and Zagier. Definition 9.2.3. Let P((0, 1)) = PQ((0, 1)) R C∞((0, 1)) be the real vector space spanned by the space of rational periods. Elements of P((0, 1)) will be called periods. 9.3. Now we are ready to state the theorem on the small ε asymptotic expansions of the functions wγ(P (ε, L),I)(a). We will regard the functional wγ(P (ε, L),I)(a) as a function of the three variables ε, L and a C∞(M). It is an element of the space of functionals O(C∞(M ),C∞((0, 1)ε) C∞((0, ∞)L))). The subscripts ε and L indicate the coordinates on the intervales (0, 1) and (0, ∞). If we fix ε but allow L and a to vary, we get a functional wγ(P (ε, L),I) O(C∞(M ),C∞((0, ∞)L)). This is a topological vector space we are interested in the behaviour of wγ(P (ε, L),I) as ε 0. The following theorem describes the small ε behaviour of wγ(P (ε, L),I). Theorem 9.3.1. Let I Oloc(C∞(M ))[[ ]] be a local functional, and let γ be a connected stable graph. (1) There exists a small ε asymptotic expansion wγ(P (ε, L),I) i=0 gi(ε)Ψi where the gi P((0, 1)ε) are periods, and Ψi O(C∞(M ),C∞((0, ∞)L)). The precise meaning of “asymptotic expansion” is as follows: there is a non-decreasing sequence dR Z, indexed by R Z 0 , such that dR as R ∞, and such that for all R, lim ε→0 ε−dR wγ(P (ε, L),I) R i=0 gi(ε)Ψi = 0.
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