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Abstract:

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants. © 2006 Elsevier Ltd. All rights reserved.

Registro:

Documento: Artículo
Título:An elementary proof of Sylvester's double sums for subresultants
Autor:D'Andrea, C.; Hong, H.; Krick, T.; Szanto, A.
Filiación:Department d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Via de les Corts Catalanes, 585, Gran, 08007, Spain
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Palabras clave:Double-sum formula; Subresultants; Vandermonde determinant
Año:2007
Volumen:42
Número:3
Página de inicio:290
Página de fin:297
DOI: http://dx.doi.org/10.1016/j.jsc.2006.09.003
Título revista:Journal of Symbolic Computation
Título revista abreviado:J. Symb. Comput.
ISSN:07477171
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_07477171_v42_n3_p290_DAndrea.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v42_n3_p290_DAndrea

Referencias:

  • Brown, W.S., Traub, J.F., On Euclid's algorithm and the theory of subresultants (1971) Journal of the ACM, 18 (4), pp. 505-514
  • Collins, G.E., Subresultants and reduced polynomial remainder sequences (1967) Journal of the ACM, 14, pp. 128-142
  • Collins, G.E., Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition (1975) Lecture Notes in Computer Science, 33, pp. 134-183. , Springer-Verlag, Berlin
  • D'Andrea, C., Krick, T., Szanto, A., Multivariate subresultants in roots (2006) Journal of Algebra, 302 (1), pp. 16-36
  • Diaz-Toca, G.M., Gonzalez-Vega, L., Various new expressions for subresultants and their applications (2004) Applicable Algebra Engineering Communication Computing, 15 (3-4), pp. 233-266
  • Gonzalez-Vega, L., A combinatorial algorithm solving some quantifier elimination problems (1996) Texts and Monographs in Symbolic Computation, , Quantifier Elimination and Cylindrical Algebraic Decomposition. Caviness B., and Johnson J. (Eds), Springer-Verlag
  • Hong, H., Ore subresultant coefficients in solutions (2001) Journal of Applicable Algebra in Engineering, Communication, and Computing, 12 (5), pp. 421-428
  • Lascoux, A., Pragacz, P., Double Sylvester sums for euclidean division, multi-Schur functions (2003) Journal of Symbolic Computation, 35, pp. 689-710
  • Lombardi, H., Roy, M.-F., El Din, M.S., New structure theorem for subresultants (2000) Journal of Symbolic Computation, 29, pp. 663-689
  • Renegar, J., On the computational complexity and geometry of the first-order theory of the reals (1992) Journal of Symbolic Computation, 13 (3), pp. 255-352
  • Sylvester, J.J., On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's function and that of the greatest algebraical common measure (1853) Trans. Roy. Soc. London, , Reprinted in: The Collected Mathematical Papers of James Joseph Sylvester, vol. 1, Chelsea Publ., New York, 1973, pp. 429-586

Citas:

---------- APA ----------
D'Andrea, C., Hong, H., Krick, T. & Szanto, A. (2007) . An elementary proof of Sylvester's double sums for subresultants. Journal of Symbolic Computation, 42(3), 290-297.
http://dx.doi.org/10.1016/j.jsc.2006.09.003
---------- CHICAGO ----------
D'Andrea, C., Hong, H., Krick, T., Szanto, A. "An elementary proof of Sylvester's double sums for subresultants" . Journal of Symbolic Computation 42, no. 3 (2007) : 290-297.
http://dx.doi.org/10.1016/j.jsc.2006.09.003
---------- MLA ----------
D'Andrea, C., Hong, H., Krick, T., Szanto, A. "An elementary proof of Sylvester's double sums for subresultants" . Journal of Symbolic Computation, vol. 42, no. 3, 2007, pp. 290-297.
http://dx.doi.org/10.1016/j.jsc.2006.09.003
---------- VANCOUVER ----------
D'Andrea, C., Hong, H., Krick, T., Szanto, A. An elementary proof of Sylvester's double sums for subresultants. J. Symb. Comput. 2007;42(3):290-297.
http://dx.doi.org/10.1016/j.jsc.2006.09.003