O papel de Sylvester no desenvolvimento da teoria dos invariantes

Magno Luiz Ferreira; Gérard Émile Grimberg

doi:10.53727/rbhc.v18i1.902

Authors

Magno Luiz Ferreira Universidade Federal do Rio de Janeiro https://orcid.org/0009-0006-0162-9910
Gérard Émile Grimberg Universidade Federal do Rio de Janeiro https://orcid.org/0000-0003-1075-5530

DOI:

https://doi.org/10.53727/rbhc.v18i1.902

Keywords:

Invariant Theory, James Joseph Sylvester, Homogeneous Polynomials

Abstract

Arthur Cayley (1821 – 1895) and James Joseph Sylvester (1814 – 1897) are seen as central characters in the development of Invariant Theory. This article aims to analyze the role of Sylvester's ideas in three-time frames: 1839 to 1847; 1849 to 1851 and 1851 to 1853, which represent the moments of change in his research perspectives. It was possible to identify the influence of the works of Julius Plücker (1801 – 1868) in the publications that supported a geometric vision of the bases of the new Theory, connections with other areas of research, such as mechanics, and the delimitation of the practices that characterized the Theory of Invariants, around the middle of the 19th century.

Downloads

Download data is not yet available.

References

Fontes primárias

ARONHOLD, S. Zur Theorie der homogenen Functionen dritten Grades von drei Variabeln. Journal für die reine und angewandte Mathematik, v. 1850, n. 40, p. 140-159, 1850. DOI: https://doi.org/10.1515/crll.1850.39.140

BOOLE, G. Exposition of a general theory of linear transformations. Cambridge Mathematical Journal, v. 3, p. 106-119, 1841.

BRIOSCHI, F. Sur l’analogie entre une classe de déterminants d’ordre pair ; et sur les déterminants binaires. Journal für die reine und angewandte Mathematik, v. 52, p. 133-141, 1856. DOI: https://doi.org/10.1515/crll.1856.52.133

BRUNO, F.F. Démonstration d’un théorème relatif à la réduction des fonctions homogènes à deux lettres à leur forme canonique. Journal de Mathématiques Pures et Appliquées, v. 17, p. 193-201, 1852.

BRUNO, F.F. Théorie des formes binaires, par le chev. F. Faà de Bruno. [S.l.]: Brero, 1876.

CAYLEY, A. Chapters in the analytical geometry of (n) dimensions. Cambridge Mathematical Journal. v. 4, p. 119-127, 1844.

CAYLEY, A. On linear transformations. Cambridge and Dublin Mathematical Journal, v. 1, p. 104-122, 1845.

CAYLEY, A. Recherches sur les matrices dont les termes sont des fonctions linéaires d’une seule indéterminée. Journal für die reine und angewandte Mathematik, v. 50, p. 313-317, 1855. DOI: https://doi.org/10.1515/crll.1855.50.313

CAYLEY, A. Mémoire sur la forme canonique des fonctions binaires. Journal für die reine und angewandte Mathematik, v. 54, p. 48-58, 1857a. DOI: https://doi.org/10.1515/crll.1857.54.48

CAYLEY, A. Addition au mémoire sur la forme canonique des fonctions binaires. Journal für die reine und angewandte Mathematik, v. 54, p. 292-292, 1857b. DOI: https://doi.org/10.1515/crll.1857.54.292

COMBESCURE, É. Sur divers points de la théorie des invariants. Journal de Mathématiques Pures et Appliquées, v. 20, p. 337-358, 1855.

HALSTED, G.B. Biography: James Joseph Sylvester. The American Mathematical Monthly, v. 4, n. 6/7, p. 159-168, 1897. DOI: https://doi.org/10.1080/00029890.1897.11998986

HERMITE, C. Extraits de lettres de m. ch. hermite à m. Jacobi sur différents objects de la théorie des nombres. Journal für die reine und angewandte Mathematik, v. 1850, n. 40, p. 261-278, 1850. DOI: https://doi.org/10.1515/crll.1850.40.261

HERMITE, C. Sur la théorie des formes quadratiques ternaires indéfinies. Journal für die reine und angewandte Mathematik, v. 1854, n. 47, p. 307-312, 1854. DOI: https://doi.org/10.1515/crll.1854.47.307

HESSE, O. Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln. Journal für die reine und angewandte Mathematik, v. 1844, n. 28, p. 68-96, 1844. DOI: https://doi.org/10.1515/crll.1844.28.68

MANSION, P. Notice sur les travaux de Jules Plücker. Bulletin des Sciences Mathématiques et Astronomiques, v. 5, p. 313-319, 1873.

PLÜCKER, M. Sur les points singuliers des courbes. Journal de Mathématiques Pures et Appliquées, v. 2, p. 11-15, 1837.

SALMON, G. Lessons introductory to the modern higher algebra. [S.l.]: Hodges, Figgis, 1885.

SPOTTISWOODE, W. Elementary theorems relating to determinants. Journal für die reine und angewandte Mathematik, v. 51, p. 209-271, 1856. DOI: https://doi.org/10.1515/crll.1856.51.209

SYLVESTER, J.J. On rational derivation from equations of coexistence, that is to say, a new and extended theory of elimination, 1839-40. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 15, p. 428-435, 1839. DOI: https://doi.org/10.1080/14786443908649916

SYLVESTER, J.J. On a linear method of eliminating between double, treble, and other systems of algebraic equations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 31, p. 425-435, 1841. DOI: https://doi.org/10.1080/14786444108650329

SYLVESTER, J.J. An account of a discovery in the theory of numbers relative to the equation Ax3 + By3 + Cz3 = Dxyz. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 31, n. 207, p. 189-191, 1847a. DOI: https://doi.org/10.1080/14786444708645823

SYLVESTER, J.J. On the equation in numbers Ax3 + By3 + cz3 = dxyz, and its associate system of equations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 31, p. 293-296, 1847b. DOI: https://doi.org/10.1080/14786444708562644

SYLVESTER, J.J. On the general solution, in certain cases, of the equation x3 + y3 + z3 = Mxyz. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 31, p. 467-471, 1847c. DOI: https://doi.org/10.1080/14786444708645893

SYLVESTER, J.J. On the intersections, contacts, and other correlations of two conics expressed by indeterminate coordinates. Cambridge and Dublin Mathematical Journal, v. 5, n. 262-282, p. 162, 1850a.

SYLVESTER, J.J. On a new class of theorems in elimination between quadratic functions. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 37, n. 249, p. 213-218, 1850b. DOI: https://doi.org/10.1080/14786445008646585

SYLVESTER, J.J. Sketch of a memoir on elimination, transformation, and canonical forms. Cambridge and Dublin Mathematical Journal, v. 6, p. 186-200, 1851a.

SYLVESTER, J.J. On the general theory of associated algebraical forms. Cambridge and Dublin Mathematical Journal, v. 6, p. 289-293, 1851b.

SYLVESTER, J.J. A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 4, n. 23, p. 138-142, 1852a. DOI: https://doi.org/10.1080/14786445208647087

SYLVESTER, J.J. On the principles of the calculus of forms (part 1). Cambridge and Dublin Mathematical Journal, v. 7, p. 52-97, 1852b.

SYLVESTER, J.J. On the principles of the calculus of forms (part 2). Cambridge and Dublin Mathematical Journal, v. 7, p. 179-217, 1852c.

SYLVESTER, J.J. Note on the calculus of forms. Cambridge and Dublin Mathematical Journal, v. 8, p. 62-64, 1853a.

SYLVESTER, J.J. On the calculus of forms, otherwise the theory of invariants. Cambridge and Dublin Mathematical Journal, v. 8, p. 256-269, 1853b.

SYLVESTER, J.J. On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. Philosophical Transactions of the Royal Society of London, n. 143, p. 407-548, 1853c. DOI: https://doi.org/10.1098/rstl.1853.0018

Fontes secundárias

BRECHENMACHER, F. Histoire du théorème de Jordan de la décomposition matricielle (1870-1930): formes de représentation et méthodes de décomposition. PhD thesis (Doctorat d’Histoire des Sciences) – École des Hautes Études en Sciences Sociales, Paris, 2006.

BRECHENMACHER, F. Lagrange and the secular equation. Lettera Matematica, v. 2, n. 1, p. 79-91, 2014. DOI: https://doi.org/10.1007/s40329-014-0051-3

CRILLY, T. The rise of Cayley’s invariant theory (1841-1862). Historia Mathematica, v. 13, n. 3, p. 241-254, 1986. DOI: https://doi.org/10.1016/0315-0860(86)90091-1

CRILLY, T. Arthur Cayley: Mathematician laureate of the Victorian Age. [S.l.]: Johns Hopkins University Press, 2006.

DIAS, L.; GRIMBERG, G. Classificação das geometrias: um diálogo entre os textos de Arthur Cayley e de Felix Klein. Revista Brasileira de História da Ciência, v. 12, n. 2, p. 207-224, 2019. DOI: https://doi.org/10.53727/rbhc.v12i2.59

PARSHALL, K.H. Toward a history of nineteenth-century invariant theory. In: Ideas and their reception. [S.l.]: Elsevier, 1989. p. 155-206. DOI: https://doi.org/10.1016/B978-0-12-599661-7.50013-6

PARSHALL, K.H. James Joseph Sylvester: Life and work in letters. [S.l.]: Oxford University Press, 1998. DOI: https://doi.org/10.1093/oso/9780198503910.001.0001

PARSHALL, K.H. James Joseph Sylvester: Jewish mathematician in a Victorian World. [S.l.]: JHU Press, 2006a. DOI: https://doi.org/10.56021/9780801882913

PARSHALL, K.H. The British development of the theory of invariants (1841-1895). BSHM Bulletin, v. 21, n. 3, p. 186-199, 2006b. DOI: https://doi.org/10.1080/17498430600964417

SINACEUR, H. Corps et modèles: essai sur l’histoire de l’algèbre réelle. [S.l.]: Vrin, 1991.