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By A. Jaffe (Chief Editor)

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Modular invariants and exponents of graphs . . . . . . . . . . . Discussion and consequences . . . . . . . . . . . . . . . . . Other Applications . . . . . . . . . . . . . . . . . . . . . Inclusions of extended U (1) theories . . . . . . . . . . . . . . Minimal models . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . .

See Polya and Szeg¨o [5] V, Exercise 145. 3 (Ruelle). Let K, L be closed subsets of the complex plane C, which do not contain 0. Suppose that the complex polynomial α + βz1 + γz2 + δz1 z2 can vanish only when z1 ∈ K or z2 ∈ L. Then the polynomial obtained by the Asano contraction, namely α + δz can vanish only when z ∈ −K ∗L, where we have written K ∗L = {z z : z ∈ K, z ∈ L}. Proof. For a proof see [6]. 4. Let the coefficients of the polynomials Qλ (z) of order ≤ N tend to the coefficients of Q(z) when λ → 0.

Let K, L be closed subsets of the complex plane C, which do not contain 0. Suppose that the complex polynomial α + βz1 + γz2 + δz1 z2 can vanish only when z1 ∈ K or z2 ∈ L. Then the polynomial obtained by the Asano contraction, namely α + δz can vanish only when z ∈ −K ∗L, where we have written K ∗L = {z z : z ∈ K, z ∈ L}. Proof. For a proof see [6]. 4. Let the coefficients of the polynomials Qλ (z) of order ≤ N tend to the coefficients of Q(z) when λ → 0. If the roots of Qλ are in the closed set K ⊂ C and if Q does not vanish identically, then the roots of Q are in K.

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Communications in Mathematical Physics - Volume 200 by A. Jaffe (Chief Editor)


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