Computations associated with the resonance arrangement

Abstract

The resonance arrangement An is the arrangement of hyperplanes in R n 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 t n−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 t n−4 coefficient of the characteristic polynomial.

Sizing the White Whale

“…If the aim is to count but not to generate the vertices of H + ∞ (d, 1), the approach proposed by Kamiya, Takemura, and Terao [15] can be applied. It was enhanced by Chroman and Singhal [6] who determined the characteristic polynomial of the 9-dimensional resonance arrangement R 9 . In addition, a formula for Betti numbers b 2 (R d ) and b 3 (R d ) has been given by Kühne [16], and a formula for b 4 (R d ) by Chroman and Singhal [6].…”

“…It was enhanced by Chroman and Singhal [6] who determined the characteristic polynomial of the 9-dimensional resonance arrangement R 9 . In addition, a formula for Betti numbers b 2 (R d ) and b 3 (R d ) has been given by Kühne [16], and a formula for b 4 (R d ) by Chroman and Singhal [6]. Pursuing the characteristic polynomial approach, Brysiewicz, Eble, and Kühne [5] computed the Betti numbers for a number of hyperplane arrangements with large symmetry groups and, independently and concurrently confirmed the value of a (9).…”

“…We establish additional combinatorial properties of a highly structured zonotope-the White Whale [4]-that allow for a significant reduction of the number of such linear optimization oracle calls, and thus to perform the orbitwise generation of all the 1 955 230 985 997 140 vertices of the 9-dimensional White Whale. This zonotope appears in a number of contexts as for example algebraic geometry, mathematical physics, economics, psychometrics, and representation theory [5,6,10,14,15,16,19,20] and is a special case of the primitive zonotopes, a family of zonotopes originally considered in relation with the question of how large the diameter of a lattice polytope can be [7]. We refer to Fukuda [11], Grünbaum [13], and Ziegler [21] for polytopes and, in particular, zonotopes.…”