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Symmetry-Based Approach to the Problem of a Perfect Cuboid - Journal of Mathematical Sciences

  • ️Sharipov, R. A.
  • ️Tue Nov 24 2020

References

  1. A. Beauville, A tale of two surfaces, e-print arXiv:1303.1910.

  2. E. Briand, “When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?” Beitr. Alg. Geom., 45, No. 2, 353–368 (2004).

    MathSciNet  MATH  Google Scholar 

  3. E. Briand andM. H. Rosas, “Milne’s volume function and vector symmetric polynomials,” J. Symbol. Comput., 44, No. 5, 583–590 (2009).

    Article  MathSciNet  Google Scholar 

  4. A. Cayley, “On the symmetric functions of the roots of certain systems of two equations,” Philos. Trans. Roy. Soc. London, 147, 717–726 (1857).

    Article  Google Scholar 

  5. E. Z. Chein, “On the derived cuboid of an Eulerian triple,” Can. Math. Bull., 20, No. 4, 509–510 (1977).

    Article  MathSciNet  Google Scholar 

  6. W. J. A. Colman, “On certain semiperfect cuboids,” Fibonacci Quart., 26, No. 1, 54–57 (1988).

    MathSciNet  MATH  Google Scholar 

  7. W. J. A. Colman, “Some observations on the classical cuboid and its parametric solutions,” Fibonacci Quart., 26, No. 4, 338–343 (1988).

    MathSciNet  MATH  Google Scholar 

  8. D. A. Cox, J. B. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, New York (1992).

    Book  Google Scholar 

  9. J. Dalbec, Geometry and combinatorics of Chow forms, Ph.D. thesis, Cornell Univ. (1995).

    MATH  Google Scholar 

  10. J. Dalbec, “Multisymmetric functions,” Beitr. Alg. Geom., 40, No. 1, 27-51 (1999).

    MathSciNet  MATH  Google Scholar 

  11. E. Freitag and R. S. Manni, Parametrization of the box variety by theta functions, e-print arXiv:1303.6495.

  12. L. González-Vega and G. Trujillo, “Multivariate Sturm–Habicht sequences: real root counting on n-rectangles and triangles,” Rev. Mat. Comput., 10, 119-130 (1997).

    MathSciNet  MATH  Google Scholar 

  13. R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York (1994).

    Book  Google Scholar 

  14. P. Halcke, Deliciae mathematicae oder mathematisches Sinnen-Confect, N. Sauer, Hamburg (1719).

  15. R. Hartshorne and R. Van Luijk, Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces, e-print arXiv:math.NT/0606700.

  16. F. Junker, “¨Uber symmetrische Functionen von mehreren Veränderlishen,” Math. Ann., 43, 225–270 (1893).

    Article  MathSciNet  Google Scholar 

  17. I. Korec, “Nonexistence of small perfect rational cuboid,” Acta Math. Univ. Comenianae, 42/43, 73–86 (1983).

  18. I. Korec, “Nonexistence of small perfect rational cuboid, II,” Acta Math. Univ. Comenianae, 44/45, 39–48 (1984).

  19. I. Korec, “Lower bounds for perfect rational cuboids,” Math. Slovaca, 42, No. 5, 565–582 (1992).

    MathSciNet  MATH  Google Scholar 

  20. A. I. Kostrikin, Introduction to Algebra, [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  21. M. Kraitchik, “On certain rational cuboids,” Scr. Math., 11, 317–326 (1945).

    MathSciNet  MATH  Google Scholar 

  22. M. Kraitchik, Théorie des Nombres. T. 3. Analyse Diophantine et Application aux Cuboides Rationelles, Gauthier-Villars, Paris (1947).

  23. M. Kraitchik, “Sur les cuboides rationelles,” Proc. Int. Congr. Math., 2, 33–34 (1954).

    Google Scholar 

  24. J. Lagrange, “Sur le dérivé du cuboide Eulérien,” Can. Math. Bull., 22, No. 2, 239–241 (1979).

    Article  Google Scholar 

  25. M. Lal and W. J. Blundon, “Solutions of the Diophantine equations x2 + y2 = l2, y2 + z2 = m2, z2 + x2 = n2, Math. Comp., 20, No. 144–147 (1966).

  26. J. Leech, “A remark on rational cuboids,” Can. Math. Bull., 24, No. 3, 377–378 (1981).

    Article  MathSciNet  Google Scholar 

  27. A. A. Masharov and R. A. Sharipov, A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, e-print arXiv:1504.07161.

  28. R. D. Matson, Results of computer search for a perfect cuboid, e-print http://unsolvedproblems.org/S58.pdf.

  29. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford (1979).

    MATH  Google Scholar 

  30. P. A. McMahon, “Memoir on symmetric functions of the roots of systems of equations,” Philos. Trans. Roy. Soc. London, 181, 481–536 (1890).

    Article  Google Scholar 

  31. P. A. McMahon, Combinatory Analysis, Vols. 1, 2, Chelsea Publishing Company, New York (1984).

    Google Scholar 

  32. M. Meskhishvili, Perfect cuboid and congruent number equation solutions, e-print arXiv:1211.6548.

  33. M. Meskhishvili, Parametric solutions for a nearly-perfect cuboid, e-print arXiv:1502.02375.

  34. M. Meskhishvili, Diophantine equations and congruent number equation solutions, e-print arXiv:1504.04584.

  35. P. Milne, “On the solutions of a set of polynomial equations,” in: Symbolic and Numerical Computation for Artificial Intelligence. Computational Mathematics and Applications (B. R. Donald, D. Kapur, and J. L. Mundy, eds.), Academic Press, London (1992), pp. 89-101.

  36. P. Pedersen, “Calculating multidimensional symmetric functions using Jacobi’s formula,” Lect. Notes Comput. Sci., 539, 304–317 (1991).

    Article  MathSciNet  Google Scholar 

  37. H. C. Pocklington, “Some Diophantine impossibilities,” Proc. Cambridge Philos. Soc., 17, 108–121 (1912).

    MATH  Google Scholar 

  38. J. R. Ramsden, A general rational solution of an equation associated with perfect cuboids, e-print arXiv:1207.5339.

  39. J. R. Ramsden and R. A. Sharipov, Inverse problems associated with perfect cuboids, e-print arXiv:1207.6764.

  40. J. R. Ramsden and R. A. Sharipov, On singularities of the inverse problems associated with perfect cuboids, e-print arXiv:1208.1859.

  41. J. R. Ramsden and R. A. Sharipov, On two algebraic parametrizations for rational solutions of the cuboid equations, e-print arXiv:1208.2587.

  42. J. R. Ramsden and R. A. Sharipov, Two and three descent for elliptic curves associated with perfect cuboids, e-print arXiv:1303.0765.

  43. R. L. Rathbun, The integer cuboid table, e-print arXiv:1705.05929.

  44. R. L. Rathbun, Four integer parametrizations for the monoclinic Diophantine piped, e-print arXiv:1705.07734.

  45. D. R. Richman, “Explicit generators of the invariants of finite groups,” Adv. Math., 124, No. 1, 49–76 (1996).

    Article  MathSciNet  Google Scholar 

  46. T. S. Roberts, “Some constraints on the existence of a perfect cuboid,” Aust. Math. Soc. Gazette, 37, No. 1, 29–31 (2010).

    MathSciNet  MATH  Google Scholar 

  47. M. H. Rosas, “MacMahon symmetric functions, the partition lattice, and Young subgroups,” J. Combin. Theory, 96A, No. 2, 326-340 (2001).

    Article  MathSciNet  Google Scholar 

  48. G.-C. Rota and J. A. Stein, “A problem of Cayley from 1857 and how he could have solved it,” Lin. Algebra Appl., 411, 167-253 (2005).

    Article  MathSciNet  Google Scholar 

  49. N. Saunderson, Elements of Algebra, Vol. 2, Cambridge Univ Press., Cambridge (1740).

    Google Scholar 

  50. J. Sawyer and C. A. Reiter, “Perfect parallelepipeds exist,” Math. Comput., 80, 1037–1040 (2011).

    Article  MathSciNet  Google Scholar 

  51. R. A. Sharipov, “Irreducible polynomials in the problem of a perfect cuboid,” Ufim. Mat. Zh., 4, No. 1, 153–160 (2012).

    Google Scholar 

  52. R. A. Sharipov, “Asymptotic approach to the problem of a perfect cuboid,” Ufim. Mat. Zh., 7, No. 3, 100–113 (2015).

    Google Scholar 

  53. R. A. Sharipov, A note on a perfect Euler cuboid, e-print arXiv:1104.1716.

  54. R. A. Sharipov, A note on the first cuboid conjecture, e-print arXiv:1109.2534.

  55. R. A. Sharipov, A note on the second cuboid conjecture. Part I, e-print arXiv:1201.1229.

  56. R. A. Sharipov, A note on the third cuboid conjecture. Part I, e-print arXiv:1203.2567.

  57. R. A. Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, e-print arXiv:1505.00724.

  58. R. A. Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, e-print arXiv:1505.02745.

  59. R. A. Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the nonlinear region, e-print arXiv:1506.04705.

  60. R. A. Sharipov, On Walter Wyss’ no perfect cuboid paper, e-print arXiv:1704.00165.

  61. R. A. Sharipov, Perfect cuboids and multisymmetric polynomials, e-print arXiv:1205.3135.

  62. R. A. Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, e-print arXiv:1206.6769.

  63. R. A. Sharipov, On the equivalence of cuboid equations and their factor equations, e-print arXiv:1207.2102.

  64. R. A. Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, e-print arXiv:1207.4081.

  65. R. A. Sharipov, On a pair of cubic equations associated with perfect cuboids, e-print arXiv:1208.0308.

  66. R. A. Sharipov, On two elliptic curves associated with perfect cuboids, e-print arXiv:1208.1227.

  67. R. A. Sharipov, A note on solutions of the cuboid factor equations, e-print arXiv:1209.0723.

  68. R. A. Sharipov, A note on rational and elliptic curves associated with the cuboid factor equations, e-print arXiv:1209.5706.

  69. L. Shläfli, “Über die Resultante eines systems mehrerer algebraishen Gleihungen,” Gesam. Math. Abhand., 2, 9–112 (1953).

    Google Scholar 

  70. B. D. Sokolowsky, A. G. VanHooft, R. M. Volkert, and C. A.Reiter, “An infinite family of perfect parallelepipeds,” Math. Comput., 83, No. 289, 2441–2454 (2014).

    Article  MathSciNet  Google Scholar 

  71. W. G. Spohn, “On the integral cuboid,” Am. Math. Mon., 79, No. 1, 57–59 (1972).

    Article  MathSciNet  Google Scholar 

  72. S. A. Stepanov, “On vector invariants of symmetric groups,” Diskr. Mat., 8, No. 2, 48–62 (1996).

    Article  MathSciNet  Google Scholar 

  73. S. A. Stepanov, “On vector invariants of symmetric groups,” Diskr. Mat., 11, No. 3, 4–14 (1999).

    Google Scholar 

  74. M. Stoll and D. Testa, The surface parametrizing cuboids, e-print arXiv:1009.0388.

  75. F. Vaccarino, The ring of multisymmetric functions e-print math.RA/0205233.

  76. R. Van Luijk, On perfect cuboids, Doctoral thesis, Math. Inst., Univ. Utrecht (2000).

  77. W. Wyss, No perfect cuboids, e-print arXiv:1506.02215.

  78. W. Wyss, On rational points on the elliptic curve E(q) : p2 + q2 = r2(1 + p2q2), e-print arXiv:1706.09842.

  79. W. Wyss, The non-commutative binomial theorem, e-print arXiv:1707.03861

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