Why Are So Many Problems Unsolved?

  • Anthony J Guttmann
Part of the Lecture Notes in Physics book series (LNP, volume 775)

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.


Unit Circle Ising Model Problem Unsolved Natural Boundary Cyclotomic Polynomial 
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  • Anthony J Guttmann

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