A note on Diophantine systems involving three symmetric polynomials

Let X¯n=(x1,…,xn) and σi(X¯n)=∑xk1…xki be i-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers a, b, c   such that for each 1⩽i⩽n1⩽i⩽n the system of Diophantine equationsσi(X¯2n)=a,σ2n−i(X¯2n)=b,σ2n(X¯2n)=c has infinitely many rational solutions. This result extends the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each k there are at least k n-tuples of integers with the same sum of i  -th powers for i=1,2,3i=1,2,3. Similar result is proved for i=1,2,4i=1,2,4 and i=−1,1,2i=−1,1,2.