On an ideal of multisymmetric polynomials associated with perfect cuboids

Abstract

A perfect Euler cuboid is a rectangular parallelepiped with integer edges, with integer face diagonals, and with integer space diagonal as well. Finding such parallelepipeds or proving their non-existence is an old unsolved mathematical problem. Algebraically the problem is described by a system of Diophantine equations. Symmetry approach to the cuboid problem is based on the natural $S_3$ symmetry of its Diophantine equations. Factorizing these equations with respect to their $S_3$ symmetry, one gets some certain ideal within the ring of multisymmetric polynomials. In the present paper this ideal is completely calculated and presented through its basis.

AmSTeX, 17 pages, amsppt style

References in corpus (2)

• A note on the third cuboid conjecture. Part I
• Perfect cuboids and multisymmetric polynomials

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• A note on rational and elliptic curves associated with the cuboid factor equations
• On two algebraic parametrizations for rational solutions of the cuboid equations
• On two elliptic curves associated with perfect cuboids
• A note on solutions of the cuboid factor equations
• Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
• Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture
• A note on invertible quadratic transformations of the real plane
• On a pair of cubic equations associated with perfect cuboids