On a class of combinatorial diophantine equations

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) withkxk = P n=1 jxij k. This is the main step to get niteness results on the number of solutions of the diophantine equation pn(x) = pm(y) if n and m have dierent parity. The combinatorial approach also allows to extend the original diophantine result to more general combinatorial situations.