The Irrational Chess Knight
نویسنده
چکیده
While in the univariate case solutions of linear recurrences with constant coeecients have rational generating functions, in the multivariate case the situation is much more interesting: even though initial conditions have rational generating functions, the corresponding solutions can have generating functions which are algebraic but not rational, and perhaps even non-algebraic. We start by an existence and uniqueness theorem for partial recurrences of the form a n = (a n+h 1 ; a n+h 2 ; : : : ; a n+h k) ; for n s ; where the values of a n for n 6 6 s are given explicitly. In particular, we show that the lattice points in the rst orthant can be enumerated in such a way that for all n the points n + h i precede n in this enumeration, if and only if the convex hull of the set H = fh 1 ; h 2 ; : : : ; h k g does not intersect the rst orthant. This condition on H which ensures existence and uniqueness of solution is assumed to be satissed in the sequel. For linear partial recurrences with constant coeecients we show that when initial conditions grow at most exponentially, the same is true of the solution, which is consequently analytic in a neighbourhood of the origin. Next we consider the algebraic nature of the generating function of the solution of such recurrences, and deene the apex of H as the componentwise maximum of the points in H f0g. When the initial conditions have rational generating functions and the apex of H is 0, the generating function of the solution is rational and is given by an explicit formula. When the initial conditions have algebraic generating functions and the apex of H has at most one positive coordinate, the generating function of the solution is algebraic and can be found by solving an algebraic equation and a system of linear equations.
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تاریخ انتشار 1998