The multiplicative anomaly of three or more commuting elliptic operators

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Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly Mₙ(A₁,...,Aₙ):=ζ(∏ᵢ₌₁ⁿ Aᵢ)∏ᵢ₌₁ⁿ ζ(Aᵢ) attached to n (suitable) operators A₁,...,Aₙ. We show that if the Aᵢ are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely Mₙ(A₁,...,Aₙ)ᵐ¹⁺...⁺ᵐⁿ =∏₁≤ ᵢ<ⱼ≤ ₙM₂(Aᵢ,Aⱼ)ᵐⁱ⁺ᵐʲ, where mⱼ is the order of Aⱼ. The proof relies on Wodzicki's 1987 formula for the pairwise multiplicative anomaly M₂(A,B) of two commuting elliptic operators.