Haruspicy and anisotropic generating functions

  • Authors:

  • Andrew Rechnitzer

  • Affiliations:

  • Department of Mathematics and Statistics, the University of Melbourne, Parkville, VIC 3010, Australia

  • Venue:
  • Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
  • Year:
  • 2003

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Abstract

Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344-347] proposed the examination of anisotropic generating functions as a test of the solvability of models of bond animals. In this article we describe a technique for examining some properties of anisotropic generating functions. For a wide range of solved and unsolved families of bond animals, we show that the coefficients of yn is rational, the degree of its numerator is at most that of its denominator, and the denominator is a product of cyclotomic polynomials. Further we are able to find a multiplicative upper bound for these denominators which, by comparison with numerical studies [Jensen, personal communication; Jensen and Guttmann, personal communication], appears to be very tight. These facts can be used to greatly reduce the amount of computation required in generating series expansions. They also have strong and negative implications for the solvability of these problems.