Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology

Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R ( P ) : = { ( x , y ) ∈ R 2 ∣ P ( x , y ) = 0 } be the real planar algebraic curve implicitly defined as the vanishing set of P . We give a deterministic algorithm...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	Discrete & computational geometry Ročník 67; číslo 3; s. 631 - 697
Hlavní autori:	Diatta, Daouda Niang, Diatta, Sény, Rouillier, Fabrice, Roy, Marie-Françoise, Sagraloff, Michael
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.04.2022 Springer Nature B.V Springer Verlag
Predmet:	Algebra Algorithms Bivariate analysis Combinatorics Complexity Computation Computational Mathematics and Numerical Analysis Computer Science Integers Mathematical analysis Mathematics Mathematics and Statistics Polynomials Symbolic Computation Topology Upper bounds Exact topology computation 68W30 68Q25 Amortized bound on algebraic numbers Real algebraic curves 13P15 14Q05 14P25
ISSN:	0179-5376, 1432-0444
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Abstract	Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R ( P ) : = { ( x , y ) ∈ R 2 ∣ P ( x , y ) = 0 } be the real planar algebraic curve implicitly defined as the vanishing set of P . We give a deterministic algorithm to compute the topology of V R ( P ) in terms of a simple straight-line planar graph G that is isotopic to V R ( P ) . The upper bound on the bit complexity of our algorithm is in O ~ ( d 5 τ + d 6 ) (The expression “the complexity is in O ~ ( f ( d , τ ) ) ” with f a polynomial in d , τ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in O ( ( log d log τ ) c f ( d , τ ) ) ”); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph G yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points.
AbstractList	Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $\tau$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of $P$. We give a deterministic and certified algorithm to compute the topology of $V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound $\tilde O (d^5 \tau + d^6)$ on the bit complexity of our algorithm, which matches the current record bound for the problem of computing the topology of a planar algebraic curve However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and the returned graph $\mathcal{G}$ yields the cylindrical algebraic decomposition information of the curve. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of such polynomials that actually exploit this amortization. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all critical points. Let P∈Z[X,Y] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ, and let VR(P):={(x,y)∈R2∣P(x,y)=0} be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of VR(P) in terms of a simple straight-line planar graph G that is isotopic to VR(P). The upper bound on the bit complexity of our algorithm is in O~(d5τ+d6)(The expression “the complexity is in O~(f(d,τ))” with f a polynomial in d,τ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in O((logdlogτ)cf(d,τ))”); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph G yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points. Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R ( P ) : = { ( x , y ) ∈ R 2 ∣ P ( x , y ) = 0 } be the real planar algebraic curve implicitly defined as the vanishing set of P . We give a deterministic algorithm to compute the topology of V R ( P ) in terms of a simple straight-line planar graph G that is isotopic to V R ( P ) . The upper bound on the bit complexity of our algorithm is in O ~ ( d 5 τ + d 6 ) (The expression “the complexity is in O ~ ( f ( d , τ ) ) ” with f a polynomial in d , τ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in O ( ( log d log τ ) c f ( d , τ ) ) ”); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph G yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points.
Author	Diatta, Daouda Niang Sagraloff, Michael Diatta, Sény Roy, Marie-Françoise Rouillier, Fabrice
Author_xml	– sequence: 1 givenname: Daouda Niang surname: Diatta fullname: Diatta, Daouda Niang organization: Université Assane Seck de Ziguinchor – sequence: 2 givenname: Sény surname: Diatta fullname: Diatta, Sény organization: Université Assane Seck de Ziguinchor – sequence: 3 givenname: Fabrice surname: Rouillier fullname: Rouillier, Fabrice organization: Sorbonne Université and Université de Paris, CNRS, INRIA, IMJ-PRG – sequence: 4 givenname: Marie-Françoise orcidid: 0000-0002-6955-6491 surname: Roy fullname: Roy, Marie-Françoise email: marie-francoise.roy@univ-rennes1.fr organization: IRMAR, Université de Rennes I – sequence: 5 givenname: Michael surname: Sagraloff fullname: Sagraloff, Michael organization: HAW Landshut
BackLink	https://inria.hal.science/hal-01891417$$DView record in HAL
BookMark	eNp9kE1PwyAch4mZiXP6BTyRePJQ_VNoaY9zUWcyXw7zTCiF2aUrFVrNvr1s1Zh42ImEPA_58ZyiUWMbjdAFgWsCwG88AEtYBDGJAGhCo68jNCaMxhEwxkZoDITnUUJ5eoJOvV9D4HPIxujp1vZN6bGxDr_aetvYTSVrj22Dp_VKF05WCj_3m0I7j2VT4mnb1pWSXRWIzuJZ7z41XtrW1na1PUPHJtj6_OecoLf7u-VsHi1eHh5n00WkaM67KDNlGWulpCGGlUSlClKlTKpjRaXMCohpaYrE0DQhiuWFoZzwVOYpUJPEKqMTdDW8-y5r0bpqI91WWFmJ-XQhdndAspwwwj9JYC8HtnX2o9e-E2vbuybME3FKM8Ip5RCobKCUs947bYSquv0vu5CgFgTELrQYQosQWuxDi6-gxv_U30UHJTpIPsDNSru_VQesb4Pbksk
CitedBy_id	crossref_primary_10_1016_j_cagd_2023_102189 crossref_primary_10_1007_s10013_023_00652_0 crossref_primary_10_1016_j_jsc_2022_08_012
Cites_doi	10.1007/3-540-33099-2 10.1145/2465506.2465523 10.1145/1390768.1390778 10.1016/j.jsc.2017.12.001 10.1016/j.cagd.2008.06.009 10.1016/j.jsc.2015.03.004 10.1007/978-3-540-73843-5_1 10.1145/1277548.1277570 10.1016/j.tcs.2013.04.014 10.1109/SYNASC.2011.17 10.1145/2465506.2465938 10.1006/jsco.2002.0531 10.1016/j.jsc.2017.03.009 10.1016/j.cam.2014.11.031 10.1007/s11786-010-0044-3 10.1109/PG.2007.18 10.1006/jcom.1996.0032 10.1016/S0167-8396(02)00167-X 10.1145/1390768.1390783 10.1109/PG.2007.32 10.1016/j.jco.2014.08.002 10.1016/j.jco.2016.07.002 10.1016/j.jsc.2014.02.001 10.1016/j.jsc.2011.11.001
ContentType	Journal Article
Copyright	The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. licence_http://creativecommons.org/publicdomain/zero
Copyright_xml	– notice: The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 – notice: The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. – notice: licence_http://creativecommons.org/publicdomain/zero
DBID	AAYXX CITATION 3V. 7SC 7TB 7XB 88I 8AL 8AO 8FD 8FE 8FG 8FK 8G5 ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO FR3 GNUQQ GUQSH HCIFZ JQ2 K7- KR7 L6V L7M L~C L~D M0N M2O M2P M7S MBDVC P5Z P62 PADUT PHGZM PHGZT PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U 1XC VOOES
DOI	10.1007/s00454-021-00353-w
DatabaseName	CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts ProQuest Central (purchase pre-March 2016) Science Database (Alumni Edition) Computing Database (Alumni Edition) ProQuest Pharma Collection Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) ProQuest Research Library Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Essentials - QC ProQuest Central Database Suite (ProQuest) ProQuest Technology Collection ProQuest One ProQuest Central Engineering Research Database ProQuest Central Student ProQuest Research Library SciTech Collection (ProQuest) ProQuest Computer Science Collection Computer Science Database Civil Engineering Abstracts ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Research Library (ProQuest) Science Database (ProQuest) Engineering Database Research Library (Corporate) Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection Research Library China Proquest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection ProQuest Central Basic Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access)
DatabaseTitle	CrossRef Research Library Prep Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College Research Library (Alumni Edition) ProQuest Pharma Collection ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Research Library Research Library China ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection Civil Engineering Abstracts ProQuest Computing Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional Advanced Technologies & Aerospace Database ProQuest One Academic UKI Edition Materials Science & Engineering Collection Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ProQuest Central (Alumni)
DatabaseTitleList	Research Library Prep
Database_xml	– sequence: 1 dbid: BENPR name: ProQuest Central url: https://www.proquest.com/central sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics Computer Science
EISSN	1432-0444
EndPage	697
ExternalDocumentID	oai:HAL:hal-01891417v1 10_1007_s00454_021_00353_w
GroupedDBID	-52 -5D -5G -BR -DZ -EM -Y2 -~C -~X .4S .86 .DC 06D 0R~ 0VY 199 1N0 1SB 203 28- 29G 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2WC 2~H 30V 3V. 4.4 406 408 409 40D 40E 5GY 5QI 5VS 67Z 692 6NX 78A 88I 8AO 8FE 8FG 8FW 8G5 8TC 8UJ 95- 95. 95~ 96X AABHQ AACDK AAGAY AAHNG AAIAL AAJBT AAJKR AAKPC AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYOK AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDZT ABECU ABFTV ABHLI ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACGOD ACHSB ACHXU ACIHN ACIPV ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEAQA AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFEXP AFGCZ AFKRA AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AI. AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARAPS ARCSS ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN AZQEC B-. BA0 BAPOH BBWZM BDATZ BENPR BGLVJ BGNMA BPHCQ BSONS C1A CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EBD EBLON EBS EDO EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GUQSH GXS H13 HCIFZ HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ I-F I09 IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ K6V K7- KDC KOV KOW KQ8 L6V LAS LLZTM LO0 M0N M2O M2P M4Y M7S MA- MQGED N2Q N9A NB0 NDZJH NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM OK1 P19 P62 P9R PADUT PF0 PKN PQQKQ PROAC PT4 PT5 PTHSS Q2X QOK QOS R4E R89 R9I REI RHV RIG RNI RNS ROL RPX RSV RYB RZK RZZ S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TN5 TSG TSK TSV TUC TUS U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW VH1 W23 W48 WK8 YIN YLTOR Z45 Z7X Z83 Z88 Z8R Z8W Z92 ZMTXR ZWQNP ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC ADHKG AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION PHGZM PHGZT PQGLB 7SC 7TB 7XB 8AL 8FD 8FK FR3 JQ2 KR7 L7M L~C L~D MBDVC PKEHL PQEST PQUKI PRINS Q9U 1XC VOOES
ID	FETCH-LOGICAL-c397t-8fdd2eccaf1f4d1c6c06ccf6e2c3aa8b023dfb5f3651c49bf37176a9603f52c83
IEDL.DBID	RSV
ISICitedReferencesCount	4
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000755417400001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0179-5376
IngestDate	Wed Nov 05 07:37:30 EST 2025 Thu Nov 27 13:42:26 EST 2025 Sat Nov 29 02:58:49 EST 2025 Tue Nov 18 21:52:30 EST 2025 Fri Feb 21 02:47:26 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	3
Keywords	Exact topology computation 68W30 68Q25 Amortized bound on algebraic numbers Real algebraic curves 13P15 14Q05 14P25
Language	English
License	licence_http://creativecommons.org/publicdomain/zero/: http://creativecommons.org/publicdomain/zero
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c397t-8fdd2eccaf1f4d1c6c06ccf6e2c3aa8b023dfb5f3651c49bf37176a9603f52c83
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0002-6955-6491 0009-0004-2972-822X
OpenAccessLink	https://inria.hal.science/hal-01891417
PQID	2638173370
PQPubID	31658
PageCount	67
ParticipantIDs	hal_primary_oai_HAL_hal_01891417v1 proquest_journals_2638173370 crossref_citationtrail_10_1007_s00454_021_00353_w crossref_primary_10_1007_s00454_021_00353_w springer_journals_10_1007_s00454_021_00353_w
PublicationCentury	2000
PublicationDate	2022-04-01
PublicationDateYYYYMMDD	2022-04-01
PublicationDate_xml	– month: 04 year: 2022 text: 2022-04-01 day: 01
PublicationDecade	2020
PublicationPlace	New York
PublicationPlace_xml	– name: New York
PublicationTitle	Discrete & computational geometry
PublicationTitleAbbrev	Discrete Comput Geom
PublicationYear	2022
Publisher	Springer US Springer Nature B.V Springer Verlag
Publisher_xml	– name: Springer US – name: Springer Nature B.V – name: Springer Verlag
References	KerberMSagraloffMA worst-case bound for topology computation of algebraic curvesJ. Symb. Comput.2012473239258286931910.1016/j.jsc.2011.11.001 AlbertiLMourrainBWintzJTopology and arrangement computation of semi-algebraic planar curvesComput. Aided Geom. Design2008258631651246337610.1016/j.cagd.2008.06.009 Burr, M., Choi, S.W., Galehouse, B., Yap, Ch.K.: Complete subdivision algorithms. II. Isotopic meshing of singular algebraic curves. In: 21st International Symposium on Symbolic and Algebraic Computation (Linz/Hagenberg 2008), pp. 87–94. ACM, New York (2008) Bodrato, M., Zanoni, A.: Long integers and polynomial evaluation with Estrin’s scheme. In: 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (Timişoara 2011), pp. 39–46. IEEE, Los Alamitos (2011) SagraloffMMehlhornKComputing real roots of real polynomialsJ. Symb. Comput.2016734686338595110.1016/j.jsc.2015.03.004 PanVYUnivariate polynomials: nearly optimal algorithms for numerical factorization and root-findingJ. Symb. Comput.2002335701733191991110.1006/jsco.2002.0531 BouzidiYLazardSMorozGPougetMRouillierFSagraloffMSolving bivariate systems using rational univariate representationsJ. Complexity2016373475355036510.1016/j.jco.2016.07.002 KerberMSagraloffMRoot refinement for real polynomials using quadratic interval refinementJ. Comput. Appl. Math.2015280377395329606710.1016/j.cam.2014.11.031 Alberti, L., Mourrain, B.: Visualisation of implicit algebraic curves. In: 15th Pacific Conference on Computer Graphics and Applications (Maui 2007), pp. 303–312. IEEE, Los Alamitos (2007) Alberti, L., Mourrain, B.: Regularity criteria for the topology of algebraic curves and surfaces. In: Mathematics of Surfaces XII (Sheffield 2007). Lecture Notes in Computer Science, vol. 4647, pp. 1–28. Springer, Berlin–Heidelberg (2007) Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of real root refinement. In: 38th International Symposium on Symbolic and Algebraic Computation (Boston 2013), pp. 299–306. ACM, New York (2013) ChengJLazardSPeñarandaLPougetMRouillierFTsigaridasEOn the topology of real algebraic plane curvesMath. Comput. Sci.201041113137273930710.1007/s11786-010-0044-3 Mehlhorn, K., Sagraloff, M., Wang, P.: From approximate factorization to root isolation. In: 38th International Symposium on Symbolic and Algebraic Computation (Boston 2013), pp. 283–290. ACM, New York (2013) Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2006). Revised version at http://perso.univ-rennes1.fr/marie-francoise.roy BeckerRSagraloffMSharmaVYapCA near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iterationJ. Symb. Comput.2018865196372521410.1016/j.jsc.2017.03.009 Wintz, J., Mourrain, B.: A subdivision arrangement algorithm for semi-algebraic curves: an overview. In: 15th Pacific Conference on Computer Graphics and Applications (Maui 2007), pp. 449–452. IEEE, Los Alamitos (2007) Gonzalez-VegaLNeculaIEfficient topology determination of implicitly defined algebraic plane curvesComput. Aided Geom. Design2002199719743194025910.1016/S0167-8396(02)00167-X Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves. In: 20th International Symposium on Symbolic and Algebraic Computation (Waterloo 2007), pp. 151–158. ACM, New York (2007) MehlhornKSagraloffMWangPFrom approximate factorization to root isolation with application to cylindrical algebraic decompositionJ. Symb. Comput.2015663469322991910.1016/j.jsc.2014.02.001 Diatta, D.N., Mourrain, B., Ruatta, O.: On the computation of the topology of a non-reduced implicit space curve. In: 21st International Symposium on Symbolic and Algebraic Computation (Linz/Hagenberg 2008), pp. 47–54. ACM, New York (2008) González-VegaLEl KahouiMAn improved upper complexity bound for the topology computation of a real algebraic plane curveJ. Complexity1996124527544142272510.1006/jcom.1996.0032 von zur GathenJGerhardJModern Computer Algebra1999New YorkCambridge University Press0936.11069 Kerber, M.: Geometric Algorithms for Algebraic Curves and Surfaces. PhD thesis, Universität des Saarlandes (2009). https://d-nb.info/1002267331/34 KobelASagraloffMOn the complexity of computing with planar algebraic curvesJ. Complexity2015312206236330599310.1016/j.jco.2014.08.002 BerberichEEmeliyanenkoPKobelASagraloffMExact symbolic-numeric computation of planar algebraic curvesTheoret. Comput. Sci.2013491132306278310.1016/j.tcs.2013.04.014 StrzebonskiATsigaridasEUnivariate real root isolation in an extension field and applicationsJ. Symb. Comput.2019923151390734610.1016/j.jsc.2017.12.001 Y Bouzidi (353_CR8) 2016; 37 353_CR26 E Berberich (353_CR6) 2013; 491 R Becker (353_CR5) 2018; 86 J von zur Gathen (353_CR13) 1999 A Strzebonski (353_CR25) 2019; 92 VY Pan (353_CR22) 2002; 33 M Kerber (353_CR18) 2015; 280 353_CR12 353_CR11 M Kerber (353_CR17) 2012; 47 353_CR16 L González-Vega (353_CR14) 1996; 12 L Alberti (353_CR3) 2008; 25 353_CR7 M Sagraloff (353_CR24) 2016; 73 353_CR9 353_CR23 A Kobel (353_CR19) 2015; 31 353_CR2 353_CR20 353_CR1 353_CR4 L Gonzalez-Vega (353_CR15) 2002; 19 J Cheng (353_CR10) 2010; 4 K Mehlhorn (353_CR21) 2015; 66
References_xml	– reference: SagraloffMMehlhornKComputing real roots of real polynomialsJ. Symb. Comput.2016734686338595110.1016/j.jsc.2015.03.004 – reference: Mehlhorn, K., Sagraloff, M., Wang, P.: From approximate factorization to root isolation. In: 38th International Symposium on Symbolic and Algebraic Computation (Boston 2013), pp. 283–290. ACM, New York (2013) – reference: Bodrato, M., Zanoni, A.: Long integers and polynomial evaluation with Estrin’s scheme. In: 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (Timişoara 2011), pp. 39–46. IEEE, Los Alamitos (2011) – reference: StrzebonskiATsigaridasEUnivariate real root isolation in an extension field and applicationsJ. Symb. Comput.2019923151390734610.1016/j.jsc.2017.12.001 – reference: Alberti, L., Mourrain, B.: Regularity criteria for the topology of algebraic curves and surfaces. In: Mathematics of Surfaces XII (Sheffield 2007). Lecture Notes in Computer Science, vol. 4647, pp. 1–28. Springer, Berlin–Heidelberg (2007) – reference: BeckerRSagraloffMSharmaVYapCA near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iterationJ. Symb. Comput.2018865196372521410.1016/j.jsc.2017.03.009 – reference: Alberti, L., Mourrain, B.: Visualisation of implicit algebraic curves. In: 15th Pacific Conference on Computer Graphics and Applications (Maui 2007), pp. 303–312. IEEE, Los Alamitos (2007) – reference: PanVYUnivariate polynomials: nearly optimal algorithms for numerical factorization and root-findingJ. Symb. Comput.2002335701733191991110.1006/jsco.2002.0531 – reference: BerberichEEmeliyanenkoPKobelASagraloffMExact symbolic-numeric computation of planar algebraic curvesTheoret. Comput. Sci.2013491132306278310.1016/j.tcs.2013.04.014 – reference: MehlhornKSagraloffMWangPFrom approximate factorization to root isolation with application to cylindrical algebraic decompositionJ. Symb. Comput.2015663469322991910.1016/j.jsc.2014.02.001 – reference: KobelASagraloffMOn the complexity of computing with planar algebraic curvesJ. Complexity2015312206236330599310.1016/j.jco.2014.08.002 – reference: Burr, M., Choi, S.W., Galehouse, B., Yap, Ch.K.: Complete subdivision algorithms. II. Isotopic meshing of singular algebraic curves. In: 21st International Symposium on Symbolic and Algebraic Computation (Linz/Hagenberg 2008), pp. 87–94. ACM, New York (2008) – reference: Diatta, D.N., Mourrain, B., Ruatta, O.: On the computation of the topology of a non-reduced implicit space curve. In: 21st International Symposium on Symbolic and Algebraic Computation (Linz/Hagenberg 2008), pp. 47–54. ACM, New York (2008) – reference: von zur GathenJGerhardJModern Computer Algebra1999New YorkCambridge University Press0936.11069 – reference: Wintz, J., Mourrain, B.: A subdivision arrangement algorithm for semi-algebraic curves: an overview. In: 15th Pacific Conference on Computer Graphics and Applications (Maui 2007), pp. 449–452. IEEE, Los Alamitos (2007) – reference: ChengJLazardSPeñarandaLPougetMRouillierFTsigaridasEOn the topology of real algebraic plane curvesMath. Comput. Sci.201041113137273930710.1007/s11786-010-0044-3 – reference: KerberMSagraloffMRoot refinement for real polynomials using quadratic interval refinementJ. Comput. Appl. Math.2015280377395329606710.1016/j.cam.2014.11.031 – reference: Kerber, M.: Geometric Algorithms for Algebraic Curves and Surfaces. PhD thesis, Universität des Saarlandes (2009). https://d-nb.info/1002267331/34 – reference: AlbertiLMourrainBWintzJTopology and arrangement computation of semi-algebraic planar curvesComput. Aided Geom. Design2008258631651246337610.1016/j.cagd.2008.06.009 – reference: KerberMSagraloffMA worst-case bound for topology computation of algebraic curvesJ. Symb. Comput.2012473239258286931910.1016/j.jsc.2011.11.001 – reference: Gonzalez-VegaLNeculaIEfficient topology determination of implicitly defined algebraic plane curvesComput. Aided Geom. Design2002199719743194025910.1016/S0167-8396(02)00167-X – reference: Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of real root refinement. In: 38th International Symposium on Symbolic and Algebraic Computation (Boston 2013), pp. 299–306. ACM, New York (2013) – reference: Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves. In: 20th International Symposium on Symbolic and Algebraic Computation (Waterloo 2007), pp. 151–158. ACM, New York (2007) – reference: BouzidiYLazardSMorozGPougetMRouillierFSagraloffMSolving bivariate systems using rational univariate representationsJ. Complexity2016373475355036510.1016/j.jco.2016.07.002 – reference: González-VegaLEl KahouiMAn improved upper complexity bound for the topology computation of a real algebraic plane curveJ. Complexity1996124527544142272510.1006/jcom.1996.0032 – reference: Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2006). Revised version at http://perso.univ-rennes1.fr/marie-francoise.roy/ – ident: 353_CR4 doi: 10.1007/3-540-33099-2 – volume-title: Modern Computer Algebra year: 1999 ident: 353_CR13 – ident: 353_CR20 doi: 10.1145/2465506.2465523 – ident: 353_CR11 doi: 10.1145/1390768.1390778 – volume: 92 start-page: 31 year: 2019 ident: 353_CR25 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2017.12.001 – volume: 25 start-page: 631 issue: 8 year: 2008 ident: 353_CR3 publication-title: Comput. Aided Geom. Design doi: 10.1016/j.cagd.2008.06.009 – volume: 73 start-page: 46 year: 2016 ident: 353_CR24 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2015.03.004 – ident: 353_CR1 doi: 10.1007/978-3-540-73843-5_1 – ident: 353_CR12 doi: 10.1145/1277548.1277570 – ident: 353_CR16 – volume: 491 start-page: 1 year: 2013 ident: 353_CR6 publication-title: Theoret. Comput. Sci. doi: 10.1016/j.tcs.2013.04.014 – ident: 353_CR7 doi: 10.1109/SYNASC.2011.17 – ident: 353_CR23 doi: 10.1145/2465506.2465938 – volume: 33 start-page: 701 issue: 5 year: 2002 ident: 353_CR22 publication-title: J. Symb. Comput. doi: 10.1006/jsco.2002.0531 – volume: 86 start-page: 51 year: 2018 ident: 353_CR5 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2017.03.009 – volume: 280 start-page: 377 year: 2015 ident: 353_CR18 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.11.031 – volume: 4 start-page: 113 issue: 1 year: 2010 ident: 353_CR10 publication-title: Math. Comput. Sci. doi: 10.1007/s11786-010-0044-3 – ident: 353_CR26 doi: 10.1109/PG.2007.18 – volume: 12 start-page: 527 issue: 4 year: 1996 ident: 353_CR14 publication-title: J. Complexity doi: 10.1006/jcom.1996.0032 – volume: 19 start-page: 719 issue: 9 year: 2002 ident: 353_CR15 publication-title: Comput. Aided Geom. Design doi: 10.1016/S0167-8396(02)00167-X – ident: 353_CR9 doi: 10.1145/1390768.1390783 – ident: 353_CR2 doi: 10.1109/PG.2007.32 – volume: 31 start-page: 206 issue: 2 year: 2015 ident: 353_CR19 publication-title: J. Complexity doi: 10.1016/j.jco.2014.08.002 – volume: 37 start-page: 34 year: 2016 ident: 353_CR8 publication-title: J. Complexity doi: 10.1016/j.jco.2016.07.002 – volume: 66 start-page: 34 year: 2015 ident: 353_CR21 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2014.02.001 – volume: 47 start-page: 239 issue: 3 year: 2012 ident: 353_CR17 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2011.11.001
SSID	ssj0004908
Score	2.3571298
Snippet	Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R ( P ) : = { ( x , y ) ∈... Let P∈Z[X,Y] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ, and let VR(P):={(x,y)∈R2∣P(x,y)=0} be the... Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $\tau$, and let...
SourceID	hal proquest crossref springer
SourceType	Open Access Repository Aggregation Database Enrichment Source Index Database Publisher
StartPage	631
SubjectTerms	Algebra Algorithms Bivariate analysis Combinatorics Complexity Computation Computational Mathematics and Numerical Analysis Computer Science Integers Mathematical analysis Mathematics Mathematics and Statistics Polynomials Symbolic Computation Topology Upper bounds
SummonAdditionalLinks	– databaseName: Science Database (ProQuest) dbid: M2P link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1BT9swFH4abAd2gMFAKwNkTbttFnHsJM4JdQjEgVY9gMQtcuwYKlUJNKWIf4-f67TbJLhwTWwn8nv2e_Z77_sAfiojhEhUTqXOEipyrqjKlaaZMw6SsVTKyrOWXGbDoby5yUfhwq0NaZXdnug3atNovCM_jlPEkuM8i07uHyiyRmF0NVBorMFH59kwTOkaxKNVXWTuGelQ6SjCloSiGV8657HnKCYoYDCN06d_DNPaHaZF_uVz_hcm9dbnfOu9__0FNoPfSfoLRdmGD1W9A1sdpwMJS3wHPg-WOK7tVxj8QdalljjPloyayTPWMDt9JU1N-pNbjDmPNRl6UpGWqNqQ_ioeTmYNOX2czitytSBieN6F6_Ozq9MLGggYqHZuyoxKa0yMMrbMCsN0qqNUa5tWseZKydLZe2PLxPI0YVrkpeXucJgqdyjiNom15HuwXjd19Q2I62w4t5EqEy2yypbaDaKyyCROlWMuesC62S90QCdHkoxJscRV9hIrnMQKL7HiqQe_ln3uF9gcb7b-4YS6bIiw2hf9ywKfRUzmTLBsznpw0EmxCIu5LVYi7MHvTg9Wr1__5P7bo32HjRiLKXwe0AGsz6aP1SF80vPZuJ0eeVV-AYBv-J4 priority: 102 providerName: ProQuest
Title	Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology
URI	https://link.springer.com/article/10.1007/s00454-021-00353-w https://www.proquest.com/docview/2638173370 https://inria.hal.science/hal-01891417
Volume	67
WOSCitedRecordID	wos000755417400001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 1432-0444 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0004908 issn: 0179-5376 databaseCode: RSV dateStart: 19970101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3fT9swED5RuoftYQzYtDJWWYg3sBTHTuw8thWoErRE_BLiJXKcmCFVKWpKEf_9bDdJx8SQxosfEjuJ7s65s-7u-wD2ZcYYC2SEheIBZhGVWEZSYW6cgyAkFCJ3rCWnfDwWNzdRXDWFlXW1e52SdH_qptnNocVhW1Jg018UP7WgbdydsIQN5xfXq27IyPHQWVPDFqykapV5_Rkv3FHrly2G_CPS_Cs56nzO8cb7vvYLfK5iTNRbGsUmrOXFFmzU_A2o2s5b8GnUYLaW2zDqW4alEpkoFsXTybPtVza2iaYF6k3ubH75XqGxIxApkSwy1FvlvtF8igaPs0WOLpekC89f4er46HIwxBXZAlYmJJljobPMt_rURLOMqFB5oVI6zH1FpRSp8e2ZTgNNw4AoFqWamoNgKM0BiOrAV4J-g_ViWuTfAZnFGaXak2mgGM91qsxDJPeywJitT1kHSC3zRFVI5JYQY5I0GMpOeomRXuKklzx14KBZ87DE4Xhz9p5RZTPRQmgPe6eJveYRERFG-IJ0YLfWdFJt3DLxQwtZSCn3OnBYa3Z1-9-v3Pm_6T_go28bKVwN0C6sz2eP-U_4oBbz-3LWhXb_aByfd6F1wrEZR_6ZG2M78gszxsFt15n9b1fG9Yc
linkProvider	Springer Nature
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1Lb9NAEB61BQk4tFBAhBZYITjBCu_DrwNCoVClahLlEKTezHrX21aK7BKnifKn-I3sbOwEkOitB662d_3Yb3Z2PTPfB_BGGSllqFKa6DikMhWKqlRpGjvnkDAWJUnhVUv68XCYnJ2loy342dbCYFplOyf6idpUGv-Rf-ARcskJEQefrn5QVI3C6GorobGCxWmxXLgtW_3x5Isb37ecH38dH_VooypAtfO9M5pYYzg-uGVWGqYjHURa26jgWiiV5M6JGZuHVkQh0zLNrXA7nki5lb6wIdeJcP1uwx2JzGKYKshHmzrM1CvgIcgp0qQ0RTq-VM9z3VFMiMDgnaCLPxzh9gWmYf62xv0rLOu93fHe__adHsJus64m3ZUhPIKtotyHvVazgjRT2D48GKx5auvHMPiMqlI1cSt3MqomS6zRdvZIqpJ0J-cYU7_UZOhFU2qiSkO6m3g_mVXk6Ho6L8h4JTSxfALfbuUVn8JOWZXFMyCusRHCBioPtYwLm2vXiYoDEzpT5UJ2gLWjnemGfR1FQCbZmjfaIyRzCMk8QrJFB96t21ytuEduvPq1A9H6QqQN73X7GR4LWJIyyeI568Bhi5qsmazqbAOZDrxvcbc5_e9bPr-5t1dwrzce9LP-yfD0AO5zLBzxOU-HsDObXhcv4K6ezy7r6UtvRgS-3zYefwEzaFik
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1Nb9MwGH61DYTgsMEAURhgITiBtTh2EucwTd1GtWld1cOQJi6ZY8cwqUpG07XqX-PX4ddNWkBitx24JrajxM_74bwfD8B7ZYQQkUqp1ElERcoVVanSNHHGQTIWS1l41pJ-MhjIi4t0uAY_21oYTKtsdaJX1KbS-I98N4yxlxznSbBrm7SI4VFv__oHRQYpjLS2dBoLiJwW85k7vtV7J0durz-EYe_z-eExbRgGqHZ2eEKlNSbEl7DMCsN0rINYaxsXoeZKydwZNGPzyPI4YlqkueXu9BMr5_VzG4VacrfuOtxL3BkT0wmH0ddVTWbq2fAQ8BRbpjQFO75sz_e9o5gcgYE8Tmd_GMX175iS-Zu_-1eI1lu-3tb__M0ew2bjb5PuQkCewFpRbsNWy2VBGtW2DY_Olv1r66dwdoBsUzVxHj0ZVqM51m47OSVVSbqjbxhrv9Jk4MlUaqJKQ7qrPAAyqcjhzXhakPMFAcX8GXy5k1d8DhtlVRYvgLjJhnMbqDzSIilsrt0iKglM5EQ45KIDrN35TDdd2ZEcZJQt-0l7tGQOLZlHSzbrwMflnOtFT5JbR79zgFoOxHbix91-htcCJlMmWDJlHdhpEZQ1SqzOVvDpwKcWg6vb_37ky9tXewsPHAyz_sng9BU8DLGexKdC7cDGZHxTvIb7ejq5qsdvvEQRuLxrOP4CR4BhkA
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Bounds+for+Polynomials+on+Algebraic+Numbers+and+Application+to+Curve+Topology&rft.jtitle=Discrete+%26+computational+geometry&rft.au=Diatta%2C+Daouda+Niang&rft.au=Diatta%2C+S%C3%A9ny&rft.au=Rouillier%2C+Fabrice&rft.au=Roy%2C+Marie-Fran%C3%A7oise&rft.date=2022-04-01&rft.issn=0179-5376&rft.eissn=1432-0444&rft.volume=67&rft.issue=3&rft.spage=631&rft.epage=697&rft_id=info:doi/10.1007%2Fs00454-021-00353-w&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s00454_021_00353_w
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0179-5376&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0179-5376&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0179-5376&client=summon