Interesting Esoterica

Haruspicy and anisotropic generating functions

Article by Rechnitzer, Andrew
  • Published in 2003
  • Added on
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.

Links

Other information

issn
01968858
journal
Advances in Applied Mathematics
number
1-2
pages
228--257
volume
30

BibTeX entry

@article{Rechnitzer2003,
	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.},
	author = {Rechnitzer, Andrew},
	issn = 01968858,
	journal = {Advances in Applied Mathematics},
	month = {feb},
	number = {1-2},
	pages = {228--257},
	title = {Haruspicy and anisotropic generating functions},
	url = {http://www.sciencedirect.com/science/article/pii/S0196885802005341},
	volume = 30,
	year = 2003,
	urldate = {2015-11-08}
}