On the bit-size of non-radical triangular sets

Xavier Dahan

doi:10.1007/978-3-319-72453-9_19

On the bit-size of non-radical triangular sets

Xavier Dahan

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

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.

Original language	English
Title of host publication	Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings
Editors	Ilias S. Kotsireas, Johannes Blomer, Dimitris E. Simos, Temur Kutsia
Publisher	Springer Verlag
Pages	264-269
Number of pages	6
ISBN (Print)	9783319724522
DOIs	https://doi.org/10.1007/978-3-319-72453-9_19
Publication status	Published - 2017
Externally published	Yes
Event	7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017 - Vienna, Austria Duration: 2017 Nov 15 → 2017 Nov 17

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	10693 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017
Country/Territory	Austria
City	Vienna
Period	17/11/15 → 17/11/17

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-319-72453-9_19

Cite this

Dahan, X. (2017). On the bit-size of non-radical triangular sets. In I. S. Kotsireas, J. Blomer, D. E. Simos, & T. Kutsia (Eds.), Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings (pp. 264-269). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10693 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-72453-9_19

On the bit-size of non-radical triangular sets. / Dahan, Xavier.

Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings. ed. / Ilias S. Kotsireas; Johannes Blomer; Dimitris E. Simos; Temur Kutsia. Springer Verlag, 2017. p. 264-269 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10693 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Dahan, X 2017, On the bit-size of non-radical triangular sets. in IS Kotsireas, J Blomer, DE Simos & T Kutsia (eds), Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10693 LNCS, Springer Verlag, pp. 264-269, 7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017, Vienna, Austria, 17/11/15. https://doi.org/10.1007/978-3-319-72453-9_19

Dahan X. On the bit-size of non-radical triangular sets. In Kotsireas IS, Blomer J, Simos DE, Kutsia T, editors, Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings. Springer Verlag. 2017. p. 264-269. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-72453-9_19

Dahan, Xavier. / On the bit-size of non-radical triangular sets. Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings. editor / Ilias S. Kotsireas ; Johannes Blomer ; Dimitris E. Simos ; Temur Kutsia. Springer Verlag, 2017. pp. 264-269 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{f37458e6e24a4da6bdd4ef6407ad2559,

title = "On the bit-size of non-radical triangular sets",

abstract = "We present upper bounds on the bit-size of coefficients of non-radical purely lexicographical Gr{\"o}bner 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.",

author = "Xavier Dahan",

note = "Funding Information: Work supported by the JSPS grant Wakate B No. 50567518. Publisher Copyright: {\textcopyright} Springer International Publishing AG 2017.; 7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017 ; Conference date: 15-11-2017 Through 17-11-2017",

year = "2017",

doi = "10.1007/978-3-319-72453-9_19",

language = "English",

isbn = "9783319724522",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "264--269",

editor = "Kotsireas, {Ilias S.} and Johannes Blomer and Simos, {Dimitris E.} and Temur Kutsia",

booktitle = "Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings",

}

TY - GEN

T1 - On the bit-size of non-radical triangular sets

AU - Dahan, Xavier

N1 - Funding Information: Work supported by the JSPS grant Wakate B No. 50567518. Publisher Copyright: © Springer International Publishing AG 2017.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85039451697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85039451697&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-72453-9_19

DO - 10.1007/978-3-319-72453-9_19

M3 - Conference contribution

AN - SCOPUS:85039451697

SN - 9783319724522

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 264

EP - 269

BT - Mathematical Aspects of Computer and Information Sciences - 7th International Conference, MACIS 2017, Proceedings

A2 - Kotsireas, Ilias S.

A2 - Blomer, Johannes

A2 - Simos, Dimitris E.

A2 - Kutsia, Temur

PB - Springer Verlag

T2 - 7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017

Y2 - 15 November 2017 through 17 November 2017

ER -

On the bit-size of non-radical triangular sets

Abstract

Publication series

Conference

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this