Autors: Stoenchev, M. R.
Title: Symmetric diophantine systems
Keywords: Elliptic curves, Mordell-Weil group, diophantine systems

Abstract: In the present work we consider Diophantine systems, i.e. systems of polynomial equations of several variables with rational coefficients. Such a sistem is called locally trivial, if it has a nonzero p-adic rational solution for all primes p, including p = ∞, or moreover, globally trivial, if it has a nonzero rational solution in Q. From global triviality follows local triviality, but the converse is not valid by Selmer’s counterexample 3x3 +4y3 +5z3 = 0. An interesting question is (see [2]): for which classes of Diophantine systems, the notions of local and global triviality are equivalent? Systems fulfilling this condition are said to satisfy the local-to-global principle. Natural examples of symmetric Diophantine systems arise in Euclidean Geometry, in problems for integer lengths of elements of geometric figures. As a result of considering many examples, we formulate conjecture, that any symmetric Diophantine system derived from Euclidean Geometry, satisfies the local-to-global pr



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