 By Morris Kline

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5 . 29 . 25 . . , √ where i = −1 and the bar means complex conjugation. Thus, with a sufficient accuracy, we have verified that ui = uj for i = j . We now prove that the singularity is a singularity for the change of coordinates (u1 , u2 , u3 ) −→ (t 1 , t 2 , t 3 ). We recall that ∂u1 (φ0 )iα = . α ∂t (φ0 )i1 This may become infinite if (φ0 )i1 = 0 for some i. In our case u1 + u2 + u3 = 3t 1 + 1 (X) , t3 ∂X = 0, ∂t 1 ∂X = 1, ∂t 2 ∂X 3 = 3 3 ∂t t and ∂ (u1 + u2 + u3 ) = 3, ∂t 1 1 ∂ (u1 + u2 + u3 ) = 3 (X) , ∂t 2 t 1 3 ∂ (u1 + u2 + u3 ) = 3 2 (X) + 3 2 (X) .

The formula of theorem can be put easily into more familiar form using the identities |a|2 − |b|2 = 1, δ log |b|2 = 2|a|δ|a| . |a|2 − 1 Indeed, ¯ ∧ δb(λ) 1 δ b(λ) |b(λ)|2 δ log |b(λ)|2 ∧ δph b(λ) = i |a(λ)|2 |a(λ)|2 = 2δ log|a(λ)| ∧ δph b(λ). Therefore, ω0 = 1 πi +∞ −∞ ¯ δ b(λ) ∧ δb(λ) 2 = 2 |a(λ)| π +∞ −∞ δ log|a(λ)| ∧ δph b(λ) dλ. Remark 2. The formula ωn = 1 πi +∞ −∞ ¯ δ b(λ) ∧ δb(λ) n λ dλ, |a(λ)|2 n = 1, 2, . . , subject to the constrains Hk = const, k = 1, . . , n, gives Darboux coordinates for higher symplectic forms.

For details the reader is referred to . e. an invertible n × n matrix solution) of the form ∞ Y0 (z, u) = p=0 φp (u)zp zµˆ zR , (10) THE SINGULARITY OF KONTSEVICH’S SOLUTION 45 where Rαβ = 0 if µα − µβ = k > 0, k ∈ N. At z = ∞ there is a formal n × n matrix solution of (8) given by F1 (u) F2 (u) YF = I + + + · · · ezU , z z2 where Fj (u)’s are n × n matrices. It is a well known result that there exist fundamental matrix solutions with asymptotic expansion YF as z → ∞ . Let l be a generic oriented line passing through the origin.