Séminaire Lotharingien de Combinatoire, B44h (2000), 7 pp.

Peter Kirschenhofer and Oliver Pfeiffer

On a Class of Combinatorial Diophantine Equations

Abstract. We give a combinatorial proof for a second order recurrence for the polynomials pn(x), where pn(k) counts the number of integer-coordinate lattice points x = (x1,...,xn) with ||x|| = \sum_{i=1}^n |xi| <= k. This is the main step to get finiteness results on the number of solutions of the diophantine equation pn(x) = pm(y) if n and m have different parity. The combinatorial approach also allows to extend the original diophantine result to more general combinatorial situations.

Received: June 28, 2000; Accepted: December 14, 2000.

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