Abstract
We present upper bounds on the bit-size of coefficients of non-radical purely lexicographical Gröbner bases (triangular sets) in dimension zero. This extends a previous work [4], constrained to radical triangular sets; it follows the same technical steps, based on interpolation. However, key notion of height of varieties is not available for points with multiplicities; therefore the bounds obtained are thus less universal and depend on some input data. We also introduce a related family of non-monic polynomials that have smaller coefficients, and smaller bounds. It is not obvious to compute them from the initial triangular set though.
Work supported by the JSPS grant Wakate B No. 50567518.
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
References
Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28(1–2), 45–124 (1999)
Chen, C., Moreno-Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)
Dahan, X.: Size of coefficients of lexicographical Gröbner baes. In: ISSAC 2009, pp. 117–126. ACM (2009)
Dahan, X., Schost, É.: Sharp estimates for triangular sets. In: ISSAC 2004, pp. 103–110. ACM Press (2004)
Dahan, X.: GCD modulo a primary triangular set of dimension zero. In: Proceedings of ISSAC 2017, pp. 109–116. ACM, New York (2017)
Moreno Maza, M., Lemaire, F., Xie, Y.: The RegularChains library (2005)
Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms I: polynomial systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 1–39. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45084-X_1
Mantzaflaris, A., Schost, É., Tsigaridas, E.: Sparse rational univariate representation. In: ISSAC 2017, p. 8 (2017)
Mehrabi, E., Schost, É.: A softly optimal monte carlo algorithm for solving bivariate polynomial systems over the integers. J. Complex. 34, 78–128 (2016)
Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Commun. Comput. 9(5), 433–461 (1999)
Wang, D.: Elimination Methods. Texts and Monographs in Symbolic Computation. Springer, Vienna (2012). https://doi.org/10.1007/978-3-7091-6202-6
Yamashita, T., Dahan, X.: Bit-size reduction of triangular sets in two and three variables. In: SCSS 2016. EPiC Series in Computing, vol. 39, pp. 169–182 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Dahan, X. (2017). On the Bit-Size of Non-radical Triangular Sets. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-72453-9_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72452-2
Online ISBN: 978-3-319-72453-9
eBook Packages: Computer ScienceComputer Science (R0)