10. CONSTRUCTING LOCAL COUNTERTERMS 61 Denote the N th partial sum of the asymptotic expansion by ΨN(ε) = N i=0 gi(ε)Φi. Then, we can define the singular part of wγ(P (ε, L),I) simply by Singε wγ(P (ε, L),I) = Singε ΨN(ε) = N i=0 (Singε gi(ε)) Φi. This singular part is independent of N, because if N is increased the function ΨN(ε) is modified only by the addition of functions of ε which are periods and which tend to zero as ε → 0. Theorem 9.3.1 implies that Singε wγ(P (ε, L),I) has the following prop- erties. Theorem 9.5.1. Let I ∈ Oloc(C∞(M ))[[ ]] be a local functional, and let γ be a connected stable graph. (1) Sing ε wγ(P (ε, L),I) is a finite sum of the form Sing ε wγ(P (ε, L),I) = fi(ε)Φi where Φi ∈ Oloc(C∞(M ),C∞((0, ∞)L)), and fi ∈ P((0, 1)) 0 are purely singular periods. (2) The limit lim ε→0 (wγ(P (ε, L),I) − Singε wγ(P (ε, L),I)) exists in the topological vector space Oloc(C∞(M ),C∞((0, ∞)L)). (3) Each Φi appearing in the finite sum above has a small L asymptotic expansion Φi ∞ j=0 fi,j(L)Ψi,j where Ψi,j ∈ Oloc(C∞(M )) is local, and fi,j(L) is a smooth function of L ∈ (0, ∞). 10. Constructing local counterterms 10.1. The heart of the proof of theorem A is the construction of local counterterms for a local interaction I ∈ Oloc(C∞(M )). This construction is simple and inductive, without the complicated graph combinatorics of the BPHZ algorithm. The theorem on the existence of local counterterms is the following.

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