Geodesic
Symmetry-Based Approach to the Problem of a Perfect Cuboid
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 143-158.

Voir la notice de l'article provenant de la source Math-Net.Ru

A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence have also not been proved. The problem of a perfect cuboid is among the unsolved mathematical problems. The problem has a natural $S_3$-symmetry connected to the permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author's results and results of J. R. Ramsden on using the $S_3$ symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.
Mots-clés : polynomial, Diophantine equation
Keywords: perfect cuboid. 
@article{INTO_2018_152_a12,
     author = {R. A. Sharipov},
     title = {Symmetry-Based {Approach} to the {Problem} of a {Perfect} {Cuboid}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {143--158},
     publisher = {mathdoc},
     volume = {152},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_152_a12/}
}
TY  - JOUR
AU  - R. A. Sharipov
TI  - Symmetry-Based Approach to the Problem of a Perfect Cuboid
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2018
SP  - 143
EP  - 158
VL  - 152
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2018_152_a12/
LA  - ru
ID  - INTO_2018_152_a12
ER  - 
%0 Journal Article
%A R. A. Sharipov
%T Symmetry-Based Approach to the Problem of a Perfect Cuboid
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2018
%P 143-158
%V 152
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2018_152_a12/
%G ru
%F INTO_2018_152_a12
R. A. Sharipov. Symmetry-Based Approach to the Problem of a Perfect Cuboid. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 143-158. http://geodesic.mathdoc.fr/item/INTO_2018_152_a12/

[58] Kostrikin A. I., Vvedenie v algebru, Nauka, M., 1977 | MR

[59] Stepanov S. A., “O vektornykh invariantakh simmetricheskikh grupp”, Diskr. mat., 8:2 (1996), 48–62 | DOI | Zbl

[60] Stepanov S. A., “O vektornykh invariantakh simmetricheskikh grupp”, Diskr. mat., 11:3 (1999), 4–14 | Zbl

[61] Sharipov R. A., “Neprivodimye polinomy v zadache o sovershennom kuboide”, Ufim. mat. zh., 4:1 (2012), 153–160

[62] Sharipov R. A., “Asimptoticheskii podkhod k zadache o sovershennom kuboide”, Ufim. mat. zh., 7:3 (2015), 100–113

[63] Beauville A., A tale of two surfaces, arXiv: 1303.1910 [math.AG] | MR

[64] Briand E., “When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?”, Beitr. Alg. Geom., 45:2 (2004), 353–368 | MR | Zbl

[65] Briand E., Rosas M. H., “Milne's volume function and vector symmetric polynomials”, J. Symbol. Comput., 44:5 (2009), 583–590 | DOI | MR | Zbl

[66] Cayley A., “On the symmetric functions of the roots of certain systems of two equations”, Phil. Trans. Roy. Soc. London, 147 (1857), 717–726 | MR

[67] Chein E. Z., “On the derived cuboid of an Eulerian triple”, Can. Math. Bull., 20:4 (1977), 509–510 | DOI | MR | Zbl

[68] Colman W. J. A., “On certain semiperfect cuboids”, Fibonacci Quart., 26:1 (1988), 54–57 | MR | Zbl

[69] Colman W. J. A., “Some observations on the classical cuboid and its parametric solutions”, Fibonacci Quart., 26:4 (1988), 338–343 | MR | Zbl

[70] Cox D. A., Little J. B., O'Shea D., Ideals, varieties, and algorithms, Springer-Verlag, New York, 1992 | MR | Zbl

[71] Dalbec J., Geometry and combinatorics of Chow forms, Ph.D. thesis, Cornell Univ., 1995 | MR

[72] Dalbec J., “Multisymmetric functions”, Beitr. Alg. Geom., 40:1 (1999), 27-–51 | MR | Zbl

[73] Freitag E., Manni R. S., Parametrization of the box variety by theta functions, arXiv: 1303.6495 [math.AG] | MR

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

[75] Guy R. K., Unsolved problems in number theory, Springer-Verlag, New York, 1994 | MR | Zbl

[76] Halcke P., Deliciae mathematicae oder mathematisches Sinnen-Confect, N. Sauer, Hamburg, 1719

[77] Hartshorne R., Van Luijk R., Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces, arXiv: math/0606700 [math.NT] | MR

[78] Junker F., “Über symmetrische Functionen von mehreren Veränderlishen”, Math. Ann., 43 (1893), 225–270 | DOI | MR | Zbl

[79] Korec I., “Nonexistence of small perfect rational cuboid”, Acta Math. Univ. Comen., 42/43 (1983), 73–86 | MR | Zbl

[80] Korec I., “Nonexistence of small perfect rational cuboid, II”, Acta Math. Univ. Comen., 44/45 (1984), 39–48 | MR | Zbl

[81] Korec I., “Lower bounds for perfect rational cuboids”, Math. Slovaca., 42:5 (1992), 565–582 | MR | Zbl

[82] Kraitchik M., “On certain rational cuboids”, Scripta Math., 11 (1945), 317–326 | MR | Zbl

[83] Kraitchik M., Théorie des nombres. 3. Analyse Diophantine et application aux cuboides rationelles, Gauthier-Villars, Paris, 1947 | MR

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

[85] Lagrange J., “Sur le dérivé du cuboide Eulérien”, Can. Math. Bull., 22:2 (1979), 239–241 | DOI | MR

[86] Lal M., Blundon W. J., “Solutions of the Diophantine equations $x^2+y^2=l^2$, $y^2+z^2=m^2$, $z^2+x^2=n^2$”, Math. Comp., 20:144–147 (1966) | MR

[87] Leech J., “A remark on rational cuboids”, Can. Math. Bull., 24:3 (1981), 377–378 | DOI | MR | Zbl

[88] Masharov A. A., Sharipov R. A., A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, arXiv: 1504.07161 [math.NT]

[89] Matson R. D. http://unsolvedproblems.org/S58.pdf

[90] Macdonald I. G., “Symmetric functions and Hall polynomials”, Oxford Math. Monogr., Clarendon Press, Oxford, 1979 | MR | Zbl

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

[92] McMahon P. A., Combinatory analysis, Chelsea Publishing Company, New York, 1984 | MR

[93] Meskhishvili M., Perfect cuboid and congruent number equation solutions, arXiv: 1211.6548 [math.NT]

[94] Meskhishvili M., Parametric solutions for a nearly-perfect cuboid, arXiv: 1502.02375 [math.NT]

[95] Meskhishvili M., Diophantine equations and congruent number equation solutions, arXiv: 1504.04584 [math.GM]

[96] Milne P., “On the solutions of a set of polynomial equations”, Symbolic and Numerical Computation for Artificial Intelligence. Computational Mathematics and Applications, eds. Donald B. R., Kapur D., Mundy J. L., Academic Press Ltd., London, 1992, 89–-101 | MR

[97] Pedersen P., “Calculating multidimensional symmetric functions using Jacobi's formula”, Lect. Notes Comput. Sci., v. 539, Springer-Verlag, 1991, 304–317 | DOI | MR

[98] Pocklington H. C., “Some Diophantine impossibilities”, Proc. Cambridge Phil. Soc., 17 (1912), 108–121

[99] Ramsden J. R., A general rational solution of an equation associated with perfect cuboids, arXiv: 1207.5339 [math.NT]

[100] Ramsden J. R., Sharipov R. A., Inverse problems associated with perfect cuboids, arXiv: 1207.6764 [math.NT]

[101] Ramsden J. R., Sharipov R. A. https://arxiv.org/abs/1208.1859 On singularities of the inverse problems associated with perfect cuboids, arXiv: 1208.1859 [math.NT]

[102] Ramsden J. R., Sharipov R. A., On two algebraic parametrizations for rational solutions of the cuboid equations, arXiv: 1208.2587 [math.NT]

[103] Ramsden J. R., Sharipov R. A., Two and three descent for elliptic curves associated with perfect cuboids, arXiv: 1303.0765 [math.NT]

[104] Rathbun R. L., The integer cuboid table, arXiv: 1705.05929 [math.NT]

[105] Rathbun R. L., Four integer parametrizations for the monoclinic Diophantine piped, arXiv: 1705.07734 [math.HO]

[106] Richman D. R., “Explicit generators of the invariants of finite groups”, Adv. Math., 124:1 (1996), 49–76 | DOI | MR | Zbl

[107] Roberts T. S., “Some constraints on the existence of a perfect cuboid”, Austr. Math. Soc. Gazette, 37:1 (2010), 29–31 | MR | Zbl

[108] Rosas M. H., “MacMahon symmetric functions, the partition lattice, and Young subgroups”, J. Combin. Theory, 96A:2 (2001), 326-–340 | DOI | MR | Zbl

[109] Rota G.-C., Stein J. A., “A problem of Cayley from 1857 and how he could have solved it”, Lin. Algebra Appl., 411 (2005), 167–-253 | DOI | MR | Zbl

[110] Saunderson N., Elements of algebra, Cambridge Univ. Press, Cambridge, 1740

[111] Sawyer J., Reiter C. A., “Perfect parallelepipeds exist”, Math. Comp., 80 (2011), 1037–1040 | DOI | MR | Zbl

[112] Sharipov R. A., A note on a perfect Euler cuboid, arXiv: 1104.1716 [math.NT]

[113] Sharipov R. A., A note on the first cuboid conjecture, arXiv: 1109.2534 [math.NT]

[114] Sharipov R. A., A note on the second cuboid conjecture, arXiv: 1201.1229 [math.NT]

[115] Sharipov R. A., A note on the third cuboid conjecture, arXiv: 1203.2567 [math.NT]

[116] Sharipov R. A., Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv: 1505.00724 [math.NT]

[117] Sharipov R. A., Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv: 1505.02745 [math.NT]

[118] Sharipov R. A., Asymptotic estimates for roots of the cuboid characteristic equation in the nonlinear region, arXiv: 1506.04705 [math.NT]

[119] Sharipov R. A., On Walter Wyss' no perfect cuboid paper, arXiv: 1704.00165 [math.NT]

[120] Sharipov R. A., Perfect cuboids and multisymmetric polynomials, arXiv: 1205.3135 [math.NT] | MR

[121] Sharipov R. A., On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv: 1206.6769 [math.NT]

[122] Sharipov R. A., On the equivalence of cuboid equations and their factor equations, arXiv: 1207.2102 [math.NT]

[123] Sharipov R. A., A biquadratic Diophantine equation associated with perfect cuboids, arXiv: 1207.4081 [math.NT]

[124] Sharipov R. A., On a pair of cubic equations associated with perfect cuboids, arXiv: 1208.0308 [math.NT]

[125] Sharipov R. A., On two elliptic curves associated with perfect cuboids, arXiv: 1208.1227 [math.NT]

[126] Sharipov R. A., A note on solutions of the cuboid factor equations, arXiv: 1209.0723 [math.NT]

[127] Sharipov R. A., A note on rational and elliptic curves associated with the cuboid factor equations, arXiv: 1209.5706 [math.NT]

[128] Shläfli L., “Über die Resultante eines systems mehrerer algebraishen Gleihungen”, Gesamm. Math. Abhandlungen., 2 (1953), 9–112, Birkhäuser Verlag

[129] Sokolowsky B. D., VanHooft A. G., Volkert R. M., Reiter C. A., “An infinite family of perfect parallelepipeds”, Math. Comp., 83:289 (2014), 2441–2454 | DOI | MR | Zbl

[130] Spohn W. G., “On the integral cuboid”, Am. Math. Monthly., 79:1 (1972), 57–59 | DOI | MR | Zbl

[131] Stoll M., Testa D., The surface parametrizing cuboids, arXiv: 1009.0388 [math.AG]

[132] Vaccarino F., The ring of multisymmetric functions, arXiv: math/0205233 [math.RA] | MR

[133] Van Luijk R., On perfect cuboids, Doctoraalscriptie, Math. Inst. Univ. Utrecht, Utrecht, 2000

[134] Wyss W., No perfect cuboids, arXiv: 1506.02215 [math.RA]

[135] Wyss W., On rational points on the elliptic curve $E(q): p^2+q^2=r^2\,(1 + p^2\,q^2)$, arXiv: 1706.09842 [math.GM]

[136] Wyss W., The non-commutative binomial theorem, arXiv: 1707.03861 [math.RA]