On some Diophantine systems involving symmetric polynomials

الأصل · EN

Let σᵢ(x₁,, xₙ)=∑1≤ k₁<k₂< <kᵢ≤ nxₖ₁ xₖᵢ 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 ZC,ZC2. More precisely, we prove that for each n≥ 4 and rational numbers a, b with ab≠ 0, the system of diophantine equations equation* σ₁(x₁,, xₙ)=a, σₙ(x₁,, xₙ)=b, equation* has infinitely many solutions depending on n-3 free parameters. A similar result is proved for the system equation* σᵢ(x₁,, xₙ)=a, σₙ(x₁,, xₙ)=b, equation* 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 equation* σᵢ(x₁,, xₙ)=a, σⱼ(x₁,, xₙ)=b, equation* 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.