On some Diophantine systems involving symmetric polynomials
Maciej Ulas2013, Mathematics of Computation
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Abstract
Let σ i (x 1 ,. .. , xn) = 1≤k 1 <k 2 <...<k i ≤n x k 1. .. x k i be the i-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai [8, 9]. More precisely, we prove that for each n ≥ 4 and rational numbers a, b with ab = 0, the system of diophantine equations σ 1 (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, has infinitely many solutions depending on n − 3 free parameters. A similar result is proved for the system σ i (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, with n ≥ 4 and 2 ≤ i < n. Here, a, b are rational numbers with b = 0. We also give some results concerning the general system of the form σ i (x 1 ,. .. , xn) = a, σ j (x 1 ,. .. , xn) = b, with suitably chosen rational values of a, b and i < j < n. Finally, we present some remarks on the systems involving three different symmetric polynomials.
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