Abstract
In this paper, we develop two variants of Bézout subresultant formulas for several polynomials, i.e., hybrid Bézout subresultant polynomial and non-homogeneous Bézout subresultant polynomial. Rather than simply extending the variants of Bézout subresultant formulas developed by Diaz–Toca and Gonzalez–Vega in 2004 for two polynomials to arbitrary number of polynomials, we propose a new approach to formulating two variants of the Bézout-type subresultant polynomials for a set of univariate polynomials. Experimental results show that the Bézout-type subresultant formulas behave better than other known formulas when used to compute multi-polynomial subresultants, among which the non-homogeneous Bézout-type formula shows the best performance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The delimitation lines in matrices hereinafter do not have any particular mathematical meaning and they are only for the presentation purpose.
References
Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput. 13(4), 865–877 (1984)
Asadi, M., Brandt, A., Jeffrey, D.J., Maza, M.M.: Subresultant chains using Bézout matrices. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2022. LNCS, vol. 13366, pp. 29–50. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-14788-3_3
Barnett, S.: Greatest common divisor of several polynomials. In: Mathematical Proceedings of the Cambridge, vol. 70, pp. 263–268. Cambridge University Press (1971)
Barnett, S.: Polynomials and Linear Control Systems. Marcel Dekker, Inc. (1983)
Bostan, A., D’Andrea, C., Krick, T., Szanto, A., Valdettaro, M.: Subresultants in multiple roots: an extremal case. Linear Algebra Appl. 529, 185–198 (2017)
Collins, G.E.: Subresultants and reduced polynomial remainder sequences. J. ACM 14(1), 128–142 (1967)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)
Cox, D.A., D’Andrea, C.: Subresultants and the Shape Lemma. Math. Comput. 92, 2355–2379 (2023)
Diaz-Toca, G.M., Gonzalez-Vega, L.: Barnett’s theorems about the greatest common divisor of several univariate polynomials through Bezout-like matrices. J. Symb. Comput. 34(1), 59–81 (2002)
Diaz-Toca, G.M., Gonzalez-Vega, L.: Various new expressions for subresultants and their applications. Appl. Algebr Eng. Comm. 15(3), 233–266 (2004)
Hong, H., Yang, J.: A condition for multiplicity structure of univariate polynomials. J. Symb. Comput. 104, 523–538 (2021)
Hong, H., Yang, J.: Subresultant of several univariate polynomials. arXiv preprint arXiv:2112.15370 (2021)
Hou, X., Wang, D.: Subresultants with the Bézout matrix. In: Computer Mathematics, pp. 19–28. World Scientific (2000)
Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using Dixon resultants. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 99–107 (1994)
Lascoux, A., Pragacz, P.: Double Sylvester sums for subresultants and multi-Schur functions. J. Symb. Comput. 35(6), 689–710 (2003)
Li, Y.B.: A new approach for constructing subresultants. Appl. Math. Comput. 183(1), 471–476 (2006)
Sylvester: On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraic common measure. Phil. Trans. R. Soc. Lond. 143, 407–548 (1853)
Terui, A.: Recursive polynomial remainder sequence and its subresultants. J. Algebra 320(2), 633–659 (2008)
Wang, D.: Decomposing polynomial systems into simple systems. J. Symb. Comput. 25(3), 295–314 (1998)
Wang, D.: Computing triangular systems and regular systems. J. Symb. Comput. 30(2), 221–236 (2000)
Acknowledgments
The authors wish to thank the anonymous reviewers for their helpful comments and insightful suggestions. The authors’ work was supported by National Natural Science Foundation of China (Grant No. 12261010), Natural Science Foundation of Guangxi (Grant No. 2023GXNSFBA026019) and the Natural Science Cultivation Project of GXMZU (Grant No. 2022MDKJ001).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Wang, W., Yang, J. (2023). Two Variants of Bézout Subresultants for Several Univariate Polynomials. In: Boulier, F., England, M., Kotsireas, I., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2023. Lecture Notes in Computer Science, vol 14139. Springer, Cham. https://doi.org/10.1007/978-3-031-41724-5_19
Download citation
DOI: https://doi.org/10.1007/978-3-031-41724-5_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-41723-8
Online ISBN: 978-3-031-41724-5
eBook Packages: Computer ScienceComputer Science (R0)