On the geometry of polar varieties
We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these...
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| Published in: | Applicable algebra in engineering, communication and computing Vol. 21; no. 1; pp. 33 - 83 |
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| Main Authors: | , , , , |
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| Language: | English |
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01.01.2010
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| ISSN: | 0938-1279, 1432-0622 |
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| Abstract | We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment. |
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| AbstractList | We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment. |
| Author | Heintz, Joos Giusti, Marc Bank, Bernd Schost, Eric Safey El Din, Mohab |
| Author_xml | – sequence: 1 givenname: Bernd surname: Bank fullname: Bank, Bernd email: bank@mathematik.hu-berlin.de organization: Humboldt-Universität zu Berlin, Institut für Mathematik – sequence: 2 givenname: Marc surname: Giusti fullname: Giusti, Marc organization: CNRS, École Polytechnique, Laboratoire LIX – sequence: 3 givenname: Joos surname: Heintz fullname: Heintz, Joos organization: Departamento de Computación, Universidad de Buenos Aires and CONICET, Ciudad Univ., Pab.I, Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria – sequence: 4 givenname: Mohab surname: Safey El Din fullname: Safey El Din, Mohab organization: UPMC, Univ Paris 06, INRIA, Paris-Rocquencourt, SALSA Project; LIP6; CNRS, UMR 7606, LIP6; UFR Ingéniérie 919, LIP6 Passy–Kennedy – sequence: 5 givenname: Eric surname: Schost fullname: Schost, Eric organization: Computer Science Department, Room 415, Middlesex College, University of Western Ontario |
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| Cites_doi | 10.1016/j.jco.2004.10.001 10.2307/1970783 10.1007/s10208-002-0065-7 10.1007/s11786-007-0003-9 10.1007/BF02567678 10.1006/jsco.2002.0563 10.1016/0304-3975(83)90002-6 10.1016/S0747-7171(88)80005-1 10.1007/BF01202721 10.1112/plms/s2-43.3.190 10.2307/2043058 10.1007/PL00004896 10.1112/plms/s2-43.2.127 10.2307/1970910 10.1007/978-3-642-57134-3 10.1007/BFb0071283 10.1007/978-0-387-75155-9_6 10.1090/conm/266/04298 10.1145/860854.860901 10.1006/jcom.1997.0432 10.1007/BFb0071291 10.1007/978-3-662-00493-7 10.1090/S0002-9939-1980-0548103-2 10.1017/CBO9781107360198.005 10.1007/978-3-7091-9459-1_17 |
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| Keywords | 68W30 Dual polar varieties Classic polar varieties Singularities 14Q10 Real polynomial equation solving Meagerly generic polar varieties 14P05 14Q15 Generic polar varieties 14B05 |
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| References | Mather (CR24) 1973; 98 Vogel (CR44) 1984 Severi (CR35) 1932; 4 Todd (CR42) 1937; 43 CR19 Safey El Din (CR33) 2007; 1 CR18 Heintz (CR17) 1983; 24 CR39 CR15 CR13 Safey El Din, Schost (CR34) 2004; 32 CR12 Room (CR30) 1938 CR32 Severi (CR36) 1902; 52 Spivak (CR38) 1965 CR31 Shafarevich (CR37) 1994 Castro, Giusti, Heintz, Matera, Pardo (CR11) 2003; 3 Bank, Giusti, Heintz, Pardo (CR6) 2005; 21 Basu, Pollack, Roy, Caviness, Johnson (CR8) 1998 Fulton (CR14) 1984 Grigor’ev, Vorobjov (CR16) 1988; 5 Bank, Giusti, Heintz, Pardo (CR5) 2004; 40 Kleiman (CR22) 1973; 28 Bank, Giusti, Heintz, Mbakop (CR3) 1997; 13 Bank, Giusti, Heintz, Mbakop (CR4) 2001; 238 Bruns, Vetter (CR10) 1988 CR7 Rouillier, Roy, Safey El Din (CR29) 2000; 167 CR9 Todd (CR43) 1937; 43 Navarro Aznar (CR27) 1980; 78 Piene (CR28) 1978; 11 CR20 CR41 CR40 Alzati, Ottaviani (CR1) 1992; 74 Kempf (CR21) 1973; 98 Lê, Teissier (CR23) 1981; 114 Matsumura (CR25) 1989 Aubry, Rouillier, Safey El Din (CR2) 2002; 34 Mork, Piene, Dickenstein (CR26) 2008 W. Vogel (117_CR44) 1984 B. Bank (117_CR6) 2005; 21 H. Matsumura (117_CR25) 1989 D. Grigor’ev (117_CR16) 1988; 5 F. Rouillier (117_CR29) 2000; 167 R. Piene (117_CR28) 1978; 11 J. Heintz (117_CR17) 1983; 24 W. Bruns (117_CR10) 1988 117_CR20 117_CR41 F. Severi (117_CR35) 1932; 4 117_CR40 D.T. Lê (117_CR23) 1981; 114 J.A. Todd (117_CR43) 1937; 43 S. Basu (117_CR8) 1998 H.C. Mork (117_CR26) 2008 J.A. Todd (117_CR42) 1937; 43 B. Bank (117_CR5) 2004; 40 D. Castro (117_CR11) 2003; 3 M. Safey El Din (117_CR34) 2004; 32 117_CR3 T.G. Room (117_CR30) 1938 117_CR9 117_CR19 B. Bank (117_CR4) 2001; 238 117_CR18 117_CR7 W. Fulton (117_CR14) 1984 117_CR39 A. Alzati (117_CR1) 1992; 74 P. Aubry (117_CR2) 2002; 34 117_CR15 I.R. Shafarevich (117_CR37) 1994 117_CR13 117_CR12 M. Safey El Din (117_CR33) 2007; 1 G. Kempf (117_CR21) 1973; 98 117_CR32 M. Spivak (117_CR38) 1965 117_CR31 S.L. Kleiman (117_CR22) 1973; 28 V. Navarro Aznar (117_CR27) 1980; 78 F. Severi (117_CR36) 1902; 52 J.N. Mather (117_CR24) 1973; 98 |
| References_xml | – volume: 114 start-page: 457 issue: 2 year: 1981 end-page: 491 ident: CR23 article-title: Variétés polaires locales et classes de Chern des variétés singulières publication-title: Ann. Math. – ident: CR18 – year: 1984 ident: CR44 publication-title: Lectures on results on Bezout’s theorem – volume: 52 start-page: 61 issue: 2 year: 1902 end-page: 118 ident: CR36 article-title: Sulle intersezioni delle varieta algebriche e sopra i loro caratteri e singolarita proiettive publication-title: Mem. Accad. Sci. Torino – ident: CR39 – volume: 21 start-page: 377 year: 2005 end-page: 412 ident: CR6 article-title: Generalized polar varieties: Geometry and algorithms publication-title: J. Complex. doi: 10.1016/j.jco.2004.10.001 – ident: CR12 – volume: 40 start-page: 519 year: 2004 end-page: 550 ident: CR5 article-title: Generalized polar varieties and an efficient real elimination procedure publication-title: Kybernetika – volume: 13 start-page: 5 year: 1997 end-page: 27 ident: CR3 article-title: Polar varieties, real equation solving and data structures: The hypersurface case publication-title: J. Complex. – volume: 98 start-page: 226 issue: 2 year: 1973 end-page: 245 ident: CR24 article-title: Generic projections publication-title: Ann. Math. doi: 10.2307/1970783 – volume: 3 start-page: 347 year: 2003 end-page: 420 ident: CR11 article-title: The hardness of polynomial equation solving publication-title: Found. Comput. Math. doi: 10.1007/s10208-002-0065-7 – volume: 28 start-page: 287 year: 1973 end-page: 297 ident: CR22 article-title: Transversality of the general translate publication-title: Compos. Math. – volume: 1 start-page: 177 year: 2007 end-page: 207 ident: CR33 article-title: Testing sign conditions on a multivariate polynomial and applications publication-title: Math. Comput. Sci. doi: 10.1007/s11786-007-0003-9 – start-page: 341 year: 1998 end-page: 350 ident: CR8 article-title: A new algorithm to find a point in every cell defined by a family of polynomials publication-title: Quantifier Elimination and Cylindrical Algebraic Decomposition – ident: CR40 – year: 1988 ident: CR10 publication-title: Determinantal rings. Lecture Notes in Mathematics 1327 – volume: 32 start-page: 417 year: 2004 end-page: 430 ident: CR34 article-title: Properness defects of projections and computation of at least one point in each connected component of a real algebraic set publication-title: J. Discre. and Comput. Geom. – volume: 74 start-page: 391 year: 1992 end-page: 412 ident: CR1 article-title: The theorem of Mather on generic projections in the setting of algebraic geometry publication-title: Manuscr. Math. doi: 10.1007/BF02567678 – volume: 34 start-page: 543 year: 2002 end-page: 560 ident: CR2 article-title: Real Solving for positive dimensional systems publication-title: J. Symb. Comput. doi: 10.1006/jsco.2002.0563 – volume: 24 start-page: 239 year: 1983 end-page: 277 ident: CR17 article-title: Definability and fast quantifier elimination in algebraically closed fields publication-title: Theoret. Comp. Sci. doi: 10.1016/0304-3975(83)90002-6 – volume: 11 start-page: 247 year: 1978 end-page: 276 ident: CR28 article-title: Polar classes of singular varieties publication-title: Ann. Scient. Éc. Norm. Sup. 4. Série, t. – ident: CR19 – year: 1989 ident: CR25 publication-title: Commutative Ring Theory – year: 1994 ident: CR37 publication-title: Basic Algebraic Geometry 1: Varieties in Projective Space (Transl. from the Russian by M. Reid.) – volume: 5 start-page: 37 year: 1988 end-page: 64 ident: CR16 article-title: Solving systems of polynomial inequalities in subexponential time publication-title: J. Symb. Comput. doi: 10.1016/S0747-7171(88)80005-1 – ident: CR15 – volume: 4 start-page: 268 year: 1932 end-page: 326 ident: CR35 article-title: La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica publication-title: Commentarii Math. Helvetici doi: 10.1007/BF01202721 – year: 1965 ident: CR38 publication-title: Calculus on Manifolds. A Modern Approach to Classical Theorems of Calculus – volume: 43 start-page: 190 year: 1937 end-page: 225 ident: CR43 article-title: The arithmetical invariants of algebraic loci publication-title: Proc. London Mat. Soc. doi: 10.1112/plms/s2-43.3.190 – volume: 167 start-page: 16 year: 2000 end-page: 750 ident: CR29 article-title: Finding at least one point in each connected component of a real algebraic set defined by a single equation publication-title: J. Complex. – ident: CR31 – ident: CR13 – ident: CR9 – start-page: 99 year: 2008 end-page: 115 ident: CR26 article-title: Polars of real singular plane curves publication-title: Algorithms in Algebraic Geometry – volume: 78 start-page: 143 year: 1980 end-page: 148 ident: CR27 article-title: On the Chern classes and the Euler characteristic for nonsingular complete intersections publication-title: Proc. Am. Math. Soc. doi: 10.2307/2043058 – ident: CR32 – volume: 238 start-page: 115 year: 2001 end-page: 144 ident: CR4 article-title: Polar varieties and efficient real elimination publication-title: Math. Z. doi: 10.1007/PL00004896 – volume: 43 start-page: 127 year: 1937 end-page: 138 ident: CR42 article-title: The geometrical invariants of algebraic loci publication-title: Proc. London Math. Soc doi: 10.1112/plms/s2-43.2.127 – ident: CR7 – volume: 98 start-page: 178 issue: 2 year: 1973 end-page: 185 ident: CR21 article-title: On the geometry of a theorem of Riemann publication-title: Ann. Math. doi: 10.2307/1970910 – year: 1938 ident: CR30 publication-title: The Geometry of Determinantal Loci – year: 1984 ident: CR14 publication-title: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] – ident: CR41 – ident: CR20 – ident: 117_CR32 – ident: 117_CR12 doi: 10.1007/978-3-642-57134-3 – volume: 11 start-page: 247 year: 1978 ident: 117_CR28 publication-title: Ann. Scient. Éc. Norm. Sup. 4. Série, t. – ident: 117_CR19 doi: 10.1007/BFb0071283 – volume: 98 start-page: 178 issue: 2 year: 1973 ident: 117_CR21 publication-title: Ann. Math. doi: 10.2307/1970910 – start-page: 99 volume-title: Algorithms in Algebraic Geometry year: 2008 ident: 117_CR26 doi: 10.1007/978-0-387-75155-9_6 – volume: 32 start-page: 417 year: 2004 ident: 117_CR34 publication-title: J. Discre. and Comput. Geom. – volume: 4 start-page: 268 year: 1932 ident: 117_CR35 publication-title: Commentarii Math. Helvetici doi: 10.1007/BF01202721 – volume-title: Calculus on Manifolds. A Modern Approach to Classical Theorems of Calculus year: 1965 ident: 117_CR38 – volume: 52 start-page: 61 issue: 2 year: 1902 ident: 117_CR36 publication-title: Mem. Accad. Sci. Torino – volume: 98 start-page: 226 issue: 2 year: 1973 ident: 117_CR24 publication-title: Ann. Math. doi: 10.2307/1970783 – volume: 74 start-page: 391 year: 1992 ident: 117_CR1 publication-title: Manuscr. Math. doi: 10.1007/BF02567678 – ident: 117_CR9 doi: 10.1090/conm/266/04298 – ident: 117_CR13 – ident: 117_CR20 – volume: 167 start-page: 16 year: 2000 ident: 117_CR29 publication-title: J. Complex. – volume: 43 start-page: 127 year: 1937 ident: 117_CR42 publication-title: Proc. London Math. Soc – volume-title: Commutative Ring Theory year: 1989 ident: 117_CR25 – volume: 238 start-page: 115 year: 2001 ident: 117_CR4 publication-title: Math. Z. doi: 10.1007/PL00004896 – ident: 117_CR41 – volume: 114 start-page: 457 issue: 2 year: 1981 ident: 117_CR23 publication-title: Ann. Math. – ident: 117_CR31 doi: 10.1145/860854.860901 – ident: 117_CR3 doi: 10.1006/jcom.1997.0432 – volume: 43 start-page: 190 year: 1937 ident: 117_CR43 publication-title: Proc. London Mat. Soc. – volume: 40 start-page: 519 year: 2004 ident: 117_CR5 publication-title: Kybernetika – volume: 5 start-page: 37 year: 1988 ident: 117_CR16 publication-title: J. Symb. Comput. doi: 10.1016/S0747-7171(88)80005-1 – volume-title: Determinantal rings. Lecture Notes in Mathematics 1327 year: 1988 ident: 117_CR10 – volume-title: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] year: 1984 ident: 117_CR14 – ident: 117_CR39 doi: 10.1007/BFb0071291 – volume: 28 start-page: 287 year: 1973 ident: 117_CR22 publication-title: Compos. Math. – volume: 21 start-page: 377 year: 2005 ident: 117_CR6 publication-title: J. Complex. doi: 10.1016/j.jco.2004.10.001 – volume: 1 start-page: 177 year: 2007 ident: 117_CR33 publication-title: Math. Comput. Sci. doi: 10.1007/s11786-007-0003-9 – volume: 3 start-page: 347 year: 2003 ident: 117_CR11 publication-title: Found. Comput. 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| SubjectTerms | Artificial Intelligence Computer Hardware Computer Science Symbolic and Algebraic Manipulation Theory of Computation |
| Title | On the geometry of polar varieties |
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