Haruspicy and anisotropic generating functions
The anisotropic generating function for a family of bond animals uses two variables; \(x\) to count the horizontal bonds and \(y\) to count the vertical bonds. The author shows the following results for the anisotropic generating function of some general classes of bond animals. The coefficient of \(y^n\) has the form \(P_n(x)/D_n(x)\) where \(D_n(x)\) is a cyclotomic polynomial and \(P_n(x)\) is a polynomial whose degree does not exceed the degree of \(D_n(x)\). An explicit polynomial \(B_n(x)\) is given such that \(D_n(x)\) necessarily divides \(B_n(x)\). Consequently \(P_n(x)/D_n(x)\) can be computed exactly from the first \(O(n^3)\) terms of its expansion. The author has done an excellent job of presenting his results in an accessible way and of explaining their wider context. This is a very readable paper. The results are proved by a technique which the author calls `haruspicy' in a humorous reference to the ancient art of divination by the disection of animals. On the basis of numerical evidence and of comparison with solved families the author speculates (a) that his families of bond animals are not solvable and (b) that \(B_n(x)\) is a good upper bound for \(D_n(x)\) in the sense that the degree of \(D_n(x)/B_n(x)\) is small (in some imprecise sense).
scientific article; zbMATH DE number 1956069