D-resultant and subresultants

El Kahoui, M’hammed

doi:10.1090/S0002-9939-05-07850-0

D-resultant and subresultants
HTML articles powered by AMS MathViewer

by M’hammed El Kahoui PDF

Proc. Amer. Math. Soc. 133 (2005), 2193-2199 Request permission

Abstract:

We establish a connection between the D-resultant of two polynomials $f(t)$ and $g(t)$ and the subresultant sequence of $f(t)-x$ and $g(t)-y$. This connection is used to decide in a more explicit way whether $\mathcal {K}(f(t),g(t))=\mathcal {K}(t)$ or $\mathcal {K}[f(t),g(t)]=\mathcal {K}[t]$. We also show how to extract a faithful parametrization from a given one.

References

Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 379502
W. S. Brown and J. F. Traub, On Euclid’s algorithm and the theory of subresultants, J. Assoc. Comput. Mach. 18 (1971), 505–514. MR 303684, DOI 10.1145/321662.321665
George E. Collins, Subresultants and reduced polynomial remainder sequences, J. Assoc. Comput. Mach. 14 (1967), 128–142. MR 215512, DOI 10.1145/321371.321381
Lionel Ducos, Optimizations of the subresultant algorithm, J. Pure Appl. Algebra 145 (2000), no. 2, 149–163. MR 1733249, DOI 10.1016/S0022-4049(98)00081-4
M’hammed El Kahoui, An elementary approach to subresultants theory, J. Symbolic Comput. 35 (2003), no. 3, 281–292. MR 1962796, DOI 10.1016/S0747-7171(02)00135-9
Arno van den Essen and Jie-Tai Yu, The $D$-resultant, singularities and the degree of unfaithfulness, Proc. Amer. Math. Soc. 125 (1997), no. 3, 689–695. MR 1353403, DOI 10.1090/S0002-9939-97-03639-3
L. González-Vega, H. Lombardi, T. Recio, and M.-F. Roy, Spécialisation de la suite de Sturm et sous-résultants, RAIRO Inform. Théor. Appl. 24 (1990), no. 6, 561–588 (French, with English summary). MR 1082916, DOI 10.1051/ita/1990240605611
Jaime Gutierrez, Rosario Rubio, and Jie-Tai Yu, $D$-resultant for rational functions, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2237–2246. MR 1896403, DOI 10.1090/S0002-9939-02-06331-1
Hoon Hong, Subresultants under composition, J. Symbolic Comput. 23 (1997), no. 4, 355–365. MR 1445431, DOI 10.1006/jsco.1996.0093
Thomas Lickteig and Marie-Françoise Roy, Sylvester-Habicht sequences and fast Cauchy index computation, J. Symbolic Comput. 31 (2001), no. 3, 315–341. MR 1814336, DOI 10.1006/jsco.2000.0427
Henri Lombardi, Marie-Françoise Roy, and Mohab Safey El Din, New structure theorem for subresultants, J. Symbolic Comput. 29 (2000), no. 4-5, 663–690. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769660, DOI 10.1006/jsco.1999.0322
R. Loos, Generalized polynomial remainder sequences, Computer algebra, Springer, Vienna, 1983, pp. 115–137. MR 728969
Andrzej Schinzel, Selected topics on polynomials, University of Michigan Press, Ann Arbor, Mich., 1982. MR 649775