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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