The problems discussed in this book, particularly that of counting the number of polygons and polyominoes in two dimensions, either by perimeter or area, seems so simple to state that it seems surprising that they haven't been exactly solved. The counting problem is so simple in concept that it can be fully explained to any schoolchild, yet it seems impossible to solve. In this chapter we develop what is essentially a numerical method that provides, at worst, strong evidence that a problem has no solution within a large class of functions, including algebraic, differentiably finite (D-finite) [27, 26] and at least a sub-class [7] of differentiably algebraic functions, called constructible differentiably algebraic (CDA) functions. Since many of the special functions of mathematical physics—in terms of which most known solutions are given—are differentiably finite, this exclusion renders the problem un-solvable within this class. Throughout this chapter the term D-unsolvable means that the problem has no solution within the class of D-finite functions as well as the sub-class of differentiably algebraic functions described above. In the next chapter, Rechnitzer shows how these ideas may be refined into a proof, in the case of polygons in two dimensions.
In fact, the exclusion is wider than D-finite functions, as we show that the solutions possess a natural boundary on the unit circle in an appropriately defined complex plane. This excludes not only D-finite functions, but a number of others as well—though we have no simple way to describe this excluded class.
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References
M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions, (Dover, New York) (1972).
R. J. Baxter, in Fundamental Problems in Statistical Mechanics, 5, (1981) 109–41. E. G. D. Cohen ed. (North Holl. Amsterdam).
R. J. Baxter, Hard hexagons: exact solution, J. Phys. A. 13, (1980) L61–70.
M. Bousquet-Mélou, (Private communication).
M. Bousquet-Mélou, A. J. Guttmann, W. P. Orrick and A. Rechnitzer, Inversion relations, reciprocity and polyominoes, Annals of Combinatorics, 3, (1999), 225–30.
M. Bousquet-Mélou and X. Viennot, Empilements de segments et q-énumération de polyomi-nos convexes dirigés. J. Combin. Theory Ser. A 60, (1992) 196–224.
F. Bergeron and C. Reutenauer, Combinatorial resolution of systems of differential equations III: a special class of differentiably algebraic series. Europ. J. Combinatorics, 11, (1990) 501–12.
M. Bousquet-Mélou, New enumerative results on two-dimensional directed animals, Disc. Math 180, (1998) 73–106.
C. H. Chang, The spontaneous magnetisation of a two-dimensional rectangular Ising model, Phys. Rev. 88, (1952) 1422–6.
A. R. Conway, R. Brak and A. J. Guttmann, Directed animals on two-dimensional lattices, J. Phys. A. 26, (1993), 3085–91.
A. R. Conway and A. J. Guttmann, Square Lattice Self-Avoiding Walks and Corrections to Scaling, Phys. Rev. Letts. 77, (1996) 5284–7.
A. R. Conway, A. J. Guttmann and M. P. Delest, The number of three-choice polygons, Mathl. Comput. Modelling 26, (1997) 51–8.
T. de Neef and I. G. Enting, Series expansions from the finite lattice method, J. Phys. A: Math. Gen. 10, (1977) 801–5.
D. Dhar, M. H. Phani and M. Barma, Enumeration of directed site animals on two-dimensional lattices, J. Phys. A. 15, (1982) L279–84.
I. G. Enting and A. J. Guttmann, In preparation.
I. G. Enting, Series Expansions from the Finite Lattice Method, Nucl. Phys. B (Proc. Suppl.) 47, (1996) 180–7.
P. Flajolet, Analytic models and ambiguity of context-free languages, Theor. Comp. Sci., 49, (1987), 283–309.
M. L. Glasser, D. B. Abraham and E. H. Lieb, Analytic properties of the free energy of the Ice models, J. Math. Phys, 13, (1972), 887–900.
A. J. Guttmann, Asymptotic Analysis of Power Series Expansions, in Phase Transitions and Critical Phenomena, 13, eds. C. Domb and J. Lebowitz, Academic Press, (1989) 1–234. (Available from www.ms.unimelb.edu.au/″tonyg)
A. J. Guttmann and A. R. Conway, Statistical Physics on the Eve of the Twenty-First Century. ed. M. T. Batchelor, (World Scientific), (1999).
A. J. Guttmann and I. Jensen, Fuchsian differential equation for the perimeter generating function of three-choice polygons, S éminaire Lotharingien de combinatoire, 54, 1–14.
Programs developed by B. Salvy, P. Zimmerman, F. Chyzak and colleagues at INRIA, France. Available from http://pauillac.inria.fr/algo
M. T. Jaekel and J. M. Maillard, Symmetry relations in exactly soluble models, J. Phys. A: Math. Gen. 15, (1982) 1309–25.
M. T. Jaekel and J. M. Maillard, Inverse functional relation on the Potts model, J. Phys. A: Math. Gen. 15, (1982) 2241–57.
G. S. Joyce, On the hard hexagon model and the theory of modular functions. Phil. Trans. of the Roy. Soc. Lond., 325, (1988), 643–706.
L. Lipshitz, D-finite power series. J. Algebra, 122, (1989) 353–73.
R. P. Stanley, Differentiably finite power series, Europ. J. Comb., 1, (1980), 175–88.
R. P. Stanley, Enumerative Combinatorics., 2
H. Tsukahara and T. Inami, Test of Guttmann and Enting's conjecture in the Eight Vertex Model, J. Phys. Soc. Japan 67, (1998) 1067–1070.
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Guttmann, A.J. (2009). Why Are So Many Problems Unsolved?. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_4
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