Papers › Computations associated with the resonance arrangement
Computations associated with the resonance arrangement
Zachary Chroman, Mihir Singhal
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The resonance arrangement 𝒜ₙ is the arrangement of hyperplanes in ℝⁿ given by all hyperplanes of the form ∑_(i ∈I) xᵢ = 0, where I is a nonempty subset of {1,…,n}. We consider the characteristic polynomial χ(𝒜ₙ; t) of the resonance arrangement, whose value Rₙ at -1 is of particular interest, and corresponds to counts of generalized retarded functions in quantum field theory, among other things. No formula is known for either the characteristic polynomial or Rₙ, though Rₙ has been computed up to n=8. By exploiting symmetry and using computational methods, we compute the characteristic polynomial of 𝒜₉, and thus obtain R₉. The coefficients of the characteristic polynomial are also equal to the so-called Betti numbers of the complexified hyperplane arrangement; that is, the coefficient of tⁿ⁻ⁱ is denoted by the Betti number bᵢ(𝒜ₙ). Explicit formulas are known for the Betti numbers up to b₃(𝒜ₙ). Using computational methods, we also obtain an explicit formula for b₄(𝒜ₙ), which gives the tⁿ⁻⁴ coefficient of the characteristic polynomial.
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