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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