Classical symmetric Diophantine systems.

This study delves into the examination of Diophantine systems, which are sets of polynomial equations with rational coefficients involving multiple variables. Specifically, the focus is on understanding the concept of local and global triviality within these systems. A Diophantine system is considered "locally trivial" if it possesses a non-zero solution in p-adic rational numbers for all prime numbers, including infinity. On the other hand, a system is deemed "globally trivial" if it has a non-zero rational solution in Q. While local triviality is a prerequisite for global triviality, it is important to note that the converse is not universally valid, as exemplified by Selmer's counterexample involving the equation 3x3+4y3+5z3=0. The main inquiry of this research revolves around investigating the conditions under which Diophantine systems exhibit equivalent notions of local and global triviality, thereby satisfying the "local-to-global principle". By analyzing various classes of Diophantine systems, the study sheds light on the existence of symmetric systems that arise in problems of Euclidean geometry, wherein the lengths of geometric elements are represented by integers. This paper is a continuous work of the [7]. [ABSTRACT FROM AUTHOR]