85:
Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set,
599:
1983 Akbulut and King introduced "Topological
Resolution Towers" as topological models of real algebraic sets, from this they obtained new topological invariants of real algebraic sets, and topologically characterized all 3-dimensional algebraic sets. These invariants later generalized by Michel
1441:
614:
1984 Benedetti and Dedo proved that not every closed smooth manifold is diffeomorphic to a totally algebraic nonsingular real algebraic set (totally algebraic means all its Z/2Z-homology cycles are represented by real algebraic
621:
1991 SchmĂŒdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz. Algebraic proof found by Wörmann. Implies
Reznick's version of Artin's theorem with uniform
1375:, A decision method for elementary algebra and geometry, Rand. Corp.. 1948; UC Press, Berkeley, 1951, Announced in : Ann. Soc. Pol. Math. 9 (1930, published 1931) 206â7; and in Fund. Math. 17 (1931) 210â239.
807:
Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise
Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006. x+662 pp.
529:
and Henry C. King gave a topological characterization of real algebraic sets with isolated singularities, and topologically characterized nonsingular real algebraic sets (not necessarily compact)
1351:, Ăber positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. ZĂŒrich 73 (1928) 141â145, in: R.P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309â313
592:
758:
399:
in 1951 and Donald Dubois in 1967 (KadisonâDubois representation theorem). Further improved by Mihai
Putinar in 1993 and Jacobi in 2001 (PutinarâJacobi representation theorem).
2011:
386:
357:
510:
introduced the "patch working" technique and used it to classify real algebraic curves of low degree. Later Ilya
Itenberg and Viro used it to produce counterexamples to the
2392:
1031:
818:
Marshall, Murray
Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp.
797:
Translated from the 1987 French original. Revised by the authors. Ergebnisse der
Mathematik und ihrer Grenzgebiete (3) , 36. Springer-Verlag, Berlin, 1998. x+430 pp.
706:
669:
557:
2091:
2078:
1879:, Michel Coste, Topologies for real algebraic geometry. Topos theoretic methods in geometry, pp. 37â100, Various Publ. Ser., 30, Aarhus Univ., Aarhus, 1979.
2017:
1493:
1784:, Local properties of analytic varieties, Differential and combinatorial topology (ed. S. Cairns), Princeton Univ. Press, Princeton N.J. (1965), 205â244.
2278:
S. Akbulut and H.C. King, All compact manifolds are homeomorphic to totally algebraic real algebraic sets, Comment. Math. Helv. 66 (1991) 139â149.
1908:(1980). "ĐŃĐžĐČŃĐ” ŃŃĐ”ĐżĐ”ĐœĐž 7, ĐșŃĐžĐČŃĐ” ŃŃĐ”ĐżĐ”ĐœĐž 8 Đž ĐłĐžĐżĐŸŃДза Đ ŃĐłŃĐŽĐ”ĐčĐ»" [Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale].
2177:
1570:
1958:
1664:
823:
812:
1710:
1065:
1018:
I. G. PetrovskiËı and O. A. OleËınik, On the topology of real algebraic surfaces, Izvestiya Akad. Nauk SSSR. Ser.Mat. 13, (1949). 389â402
2353:
E. Bierstone and P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant,
482:
128:
1891:, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. In Topology (Leningrad, 1982), volume 1060 of
1844:, "Quantifier elimination for real closed fields by cylindrical algebraic decomposition", Lect. Notes Comput. Sci. 33, 134â183, 1975
1049:, Sur lâhomologie des variÂŽetÂŽes algebriques rÂŽeelles, in: S. S. Cairns (ed.), Differential and Combinatorial Topology, pp. 255â265,
2117:
2065:
1796:, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205â224
887:
802:
788:
974:
137:
is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of
2509:
1154:
250:
158:
2265:
R. Benedetti and M. Dedo, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism,
203:
1420:
1360:
417:
297:
162:
721:
2000 Scheiderer's local-global principle and related non-strict extension of SchmĂŒdgen's positivstellensatz in dimensions †2.
410:
118:
1525:
878:. Translations of Mathematical Monographs. Vol. 88. Translated from the Russian by Smilka Zdravkovska. Providence, RI:
421:
312:
246:
87:
913:, Solution d'une question particuliĂ©re du calcul des inĂ©galitĂ©s. Bull. sci. Soc. Philomn. Paris 99â100. OEuvres 2, 315â319.
879:
618:
1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.
1892:
728:
proved that every closed smooth 3âmanifold is the real part of a compact complex manifold which can be obtained from
460:
2438:
J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry,
1063:
Basu, Saugata (1999). "On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets".
2344:
R. Benedetti and A. Marin, DĂ©chirures de variĂ©tĂ©s de dimension trois ...., Comment. Math. Helv. 67 (1992), 514â545.
1910:
1738:
G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87â97.
1236:
1050:
849:
228:
170:
73:
with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are
2417:
C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta
Mathematica 119 (2006), no. 4, 395â410.
2112:
Selman
Akbulut and Henry C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992)
2405:
1583:
1404:
1158:
285:
2318:
S. Akbulut and H.C. King On approximating submanifolds by algebraic sets and a solution to the Nash conjecture,
127:
is concerned with the algorithmic aspects of real algebraic (and semialgebraic) geometry. The main algorithm is
2439:
2354:
2319:
2253:
2103:
S. Akbulut and H.C. King, The topology of real algebraic sets, L'Enseignement MathĂ©matique 29 (1983), 221â261.
1864:
1702:
1002:
388:
which is a complete intersection (from the conclusion of this theorem the word "component" can not be removed).
70:
625:
1992 Akbulut and King proved ambient versions of the Nash-Tognoli theorem: Every closed smooth submanifold of
2172:
McCrory, Clint; ParusiĆski, Adam (2007), "Algebraically constructible functions: real algebra and topology",
406:
proved that every closed smooth manifold is diffeomorphic to a nonsingular component of a real algebraic set.
56:
74:
2266:
1876:
783:
S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992)
500:
486:
464:
2309:
B. Reznick, Uniform denominators in
Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75â97.
1332:
Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse
264:
2049:
1769:
1601:
1388:
1119:
1006:
928:
562:
731:
596:
1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.
515:
242:
1987:
362:
333:
1596:
403:
359:
with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of
2484:
304:
are triangularizable, but the necessary tools have not been developed to make the argument rigorous.
2210:
1292:
1163:
990:
511:
178:
166:
150:
1756:
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449â474.
2367:
2332:
2235:
2181:
2135:
1860:
1618:
1552:
1384:
1309:
1269:
1203:
1136:
1092:
945:
493:
442:
316:
110:
48:
36:
2426:
2379:
1363:, Topologische BegrĂŒndung des KalkĂŒls der abzĂ€hlenden Geometrie. Math. Ann. 102, 337â362 (1929).
725:
712:
320:
218:
2471:
S. Akbulut, Real algebraic structures, Proceedings of GGT, (2005) 49â58, arXiv:math/0601105v3.
2113:
1954:
1841:
1832:, Su una congettura di Nash, Annali della Scuola Normale Superiore di Pisa 27, 167â185 (1973).
1768:, Resolution of singularities of an algebraic variety over a field of characteristic zero. I,
1470:
1220:
883:
871:
819:
808:
798:
784:
608:
604:
519:
478:
474:
proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set.
301:
146:
66:
52:
2443:
2300:
T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998.
2219:
2144:
2026:
1964:
1931:
1793:
1765:
1719:
1673:
1636:
1610:
1534:
1460:
1450:
1301:
1261:
1195:
1128:
1082:
1074:
962:
937:
893:
853:
715:
showed that not every closed 3-manifold is a projective real 3-fold which is birational to
449:
431:
257:
211:
186:
102:
2291:-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203â206.
2231:
2195:
2158:
1848:
1800:
1687:
1548:
1506:
684:
647:
535:
2447:
2227:
2191:
2154:
1968:
1935:
1845:
1829:
1813:
1797:
1781:
1683:
1544:
1502:
1488:
1400:
989:, Sur le Nombre des Racines dâune Ăquation AlgĂ©brique Comprise Entre des Limites DonnĂ©es,
986:
897:
857:
471:
438:
396:
327:
182:
131:. It is used to cut semialgebraic sets into nice pieces and to compute their projections.
95:
1348:
165:.) The relation of real algebra to real algebraic geometry is similar to the relation of
2061:
2045:
1747:
S. Lang, Algebra. AddisonâWesley Publishing Co., Inc., Reading, Mass. 1965 xvii+508 pp.
1465:
1437:
1433:
1416:
1287:
910:
526:
453:
392:
174:
154:
1419:
and Henry C. King, Submanifolds and homology of nonsingular real algebraic varieties,
2503:
2239:
2149:
2130:
1582:
T. Jacobi, A representation theorem for certain partially ordered commutative rings.
1556:
1372:
1327:
1313:
1273:
1232:
1207:
1140:
1110:
308:
274:
138:
114:
40:
2048:
and Henry C. King, The topology of real algebraic sets with isolated singularities,
1096:
770:
is smoothly isotopic to the real part of a nonsingular complex algebraic subset of
232:
190:
142:
2391:
C. Scheiderer, Sums of squares of regular functions on real algebraic varieties.
2031:
1983:
1520:
1046:
17:
965:, BeitrÀge zur Theorie der linearen Ungleichungen. IV+ 76 S. Diss., Basel (1936).
766:
2005 Akbulut and King showed that not every nonsingular real algebraic subset of
1027:
923:
678:
1997 Bierstone and Milman proved a canonical resolution of singularities theorem
629:
is isotopic to the nonsingular points (component) of a real algebraic subset of
273:
1916 Fejér's conjecture about nonnegative trigonometric polynomials. (Solved by
207:
44:
28:
1442:
Proceedings of the National Academy of Sciences of the United States of America
681:
1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from
675:
is homeomorphic to a possibly singular affine real algebraic rational threefold
117:
are examples of semialgebraic mappings. Piecewise polynomial mappings (see the
1678:
1659:
1305:
1249:
281:
106:
496:
proved that every subanalytic set admits a stratification with condition (w).
1905:
1888:
977:, MĂ©moires divers prĂ©sentĂ©s par des savants Ă©trangers 6, pp. 273â318 (1835).
640:
1992 Benedetti and Marin proved that every compact closed smooth 3-manifold
507:
91:
2077:
S. Akbulut and H.C. King, Real algebraic structures on topological spaces,
1539:
1474:
1223:, šUber trigonometrische Polynome, J. Reine Angew. Math. 146 (1916), 53â82.
2492:
2174:
Arc spaces and additive invariants in real algebraic and analytic geometry
1455:
763:
2003 Welschinger introduces an invariant for counting real rational curves
441:
proved that every analytic variety admits a stratification satisfying the
2382:, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63â73
1953:. Oberwolfach Seminars. Vol. 35. Basel: BirkhĂ€user. pp. 34â35.
1087:
600:
Coste and Krzysztof Kurdyka as well as Clint McCrory and Adam ParusiĆski.
2429:, The Nash conjecture for nonprojective threefolds, arXiv:math/0009108v1
2223:
1724:
1622:
1491:(1951), "A representation theory for commutative topological algebra",
1265:
1199:
1132:
1078:
949:
2186:
1115:"Uber die Darstellung definiter Formen als Summe von Formenquadraten"
290:
1927 KrullâBaer Theorem (connection between orderings and valuations)
1922:"Curves of degree 7, curves of degree 8 and Ragsdale's conjecture".
1614:
1569:
Mihai Putinar, Positive polynomials on compact semi-algebraic sets.
1183:
1114:
941:
231:. (This bound on the number of components was later extended to all
2208:
Bröcker, Ludwig (1984). "Minimale erzeugung von Positivbereichen".
671:
by a sequence of blow ups and downs along smooth centers, and that
395:'s representation theorem for partially ordered rings. Improved by
848:. London Mathematical Society Lecture Note Series. Vol. 248.
603:
1984 Ludwig Bröcker's theorem on minimal generation of basic open
559:
is the link of a real algebraic set with isolated singularity in
1239:, Functional Analysis, Frederick Ungar Publ. Co., New York, 1955.
420:. Rediscovered and popularized by Stengle in 1974. (Krivine uses
1252:(1927). "Uber die Zerlegung definiter Funktionen in Quadrate".
2366:
G. Mikhalkin, Blow up equivalence of smooth closed manifolds,
267:
showed that not every real algebraic surface is birational to
2090:
S. Akbulut and L. Taylor, A topological resolution theorem,
1949:
Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007).
633:, and they extended this result to immersed submanifolds of
2404:
C. Scheiderer, Sums of squares on real algebraic curves,
1863:, Stratifications de Whitney et théorÚme de Bertini-Sard,
2458:
S. Akbulut and H.C. King, Transcendental submanifolds of
1639:; Pierce, Richard Scott (1956). "Lattice ordered rings".
293:
1928 PĂłlya's Theorem on positive polynomials on a simplex
2486:
The Role of Hilbert Problems in Real Algebraic Geometry
2331:
S. Akbulut and H.C. King, Algebraicity of Immersions,
235:
of all real algebraic sets and all semialgebraic sets.)
1005:Ăber Vieltheiligkeit der ebenen algebraischen Curven,
77:, i.e., mappings whose graphs are semialgebraic sets.
1990:
1772:(2) 79 (1): (1964) 109â203, and part II, pp. 205â326.
1403:, Algebraische approximation von Mannigfaltigkeiten,
793:
Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise.
734:
687:
650:
565:
538:
365:
336:
2494:
Real Algebraic and Analytic Geometry Preprint Server
2252:C. Scheiderer, Stability index of real varieties.
1330:(1927), "Ăber nicht-archimedisch geordnete Körper",
1184:"Sulla connessione delle superfizie razionali reali"
206:
for systems of linear inequalities. Rediscovered by
197:
Timeline of real algebra and real algebraic geometry
260:(Can be reformulated as linear positivstellensatz.)
2131:"On the link of a stratum in a real algebraic set"
2005:
752:
700:
663:
586:
551:
380:
351:
86:but it is always a semialgebraic set: this is the
2393:Transactions of the American Mathematical Society
503:discover the real spectrum of a commutative ring.
418:Krivine's Nullstellensatz and Positivestellensatz
1521:"A note on David Harrison's theory of preprimes"
1387:, A new decision method for elementary algebra,
1032:Proceedings of the American Mathematical Society
532:1980 Akbulut and King proved that every knot in
427:1964 Lojasiewicz triangulated semi-analytic sets
424:while Stengle uses Lang's homomorphism theorem.)
2176:, Panoramas et SynthĂšses, vol. 24, Paris:
1293:Journal fĂŒr die reine und angewandte Mathematik
1164:Journal fĂŒr die Reine und Angewandte Mathematik
991:Journal fĂŒr die reine und angewandte Mathematik
708:by a sequence of topological blow ups and downs
330:proved that every closed smooth submanifold of
1159:"Ăber die Theorie der Einfachen Ungleichungen"
760:by a sequence of real blow ups and blow downs.
224:1856 Hermite's theorem on real root counting.
8:
2064:and Henry C. King, All knots are algebraic,
2018:Journal of the American Mathematical Society
1984:"Enumerative tropical algebraic geometry in
1494:Memoirs of the American Mathematical Society
434:proved the resolution of singularity theorem
452:finds a positive polynomial which is not a
238:1888 Hilbert's theorem on ternary quartics.
2129:Coste, Michel; Kurdyka, Krzysztof (1992).
1030:, On the Betti numbers of real varieties,
926:(1919). "Systems of linear inequalities".
69:, i.e. real-number solutions to algebraic
2462:Comment. Math. Helv., 80, (2005), 427â432
2185:
2148:
2030:
1997:
1993:
1992:
1989:
1723:
1677:
1538:
1464:
1454:
1086:
744:
740:
737:
736:
733:
692:
686:
655:
649:
572:
568:
567:
564:
543:
537:
489:and allows to implement it on a computer.
372:
368:
367:
364:
343:
339:
338:
335:
300:sketches a proof that real algebraic and
149:) and their applications to the study of
109:are examples of semialgebraic sets. Real
1818:Functional Analysis and Its Applications
1641:Anais da Academia Brasileira de CiĂȘncias
1290:(1932). "Allgemeine Bewertungstheorie".
485:algorithm, which improves Tarski's real
105:are examples of real algebraic sets and
1895:, pages 187â200. Springer, Berlin, 1984
1423:, vol. 107, no. 1 (Feb., 1985) p.72
836:
607:(improved and extended to basic closed
413:formulated. (Solved in dimensions †2.)
1571:Indiana University Mathematics Journal
1188:Annali di Matematica Pura ed Applicata
846:Tame topology and o-minimal structures
1665:Rocky Mountain Journal of Mathematics
1066:Discrete & Computational Geometry
125:Computational real algebraic geometry
7:
2092:Publications MathĂ©matiques de l'IHĂS
2079:Publications MathĂ©matiques de l'IHĂS
1599:(1952). "Real algebraic manifolds".
1660:"On the PierceâBirkhoff conjecture"
483:cylindrical algebraic decomposition
173:. Related fields are the theory of
129:cylindrical algebraic decomposition
121:) are also semialgebraic mappings.
51:with real-number coefficients, and
2335:, vol. 31, no. 4, (1992), 701â712.
587:{\displaystyle \mathbb {R} ^{n+1}}
25:
2066:Commentarii Mathematici Helvetici
1438:"A general theory of spectra. I."
753:{\displaystyle \mathbb {CP} ^{3}}
2006:{\displaystyle \mathbb {R} ^{2}}
381:{\displaystyle \mathbb {R} ^{n}}
352:{\displaystyle \mathbb {R} ^{n}}
2395:352 (2000), no. 3, 1039â1069.
1421:American Journal of Mathematics
2178:Société mathématique de France
1820:, volume 6, pp. 136â138 (1972)
1711:Journal d'Analyse Mathématique
1526:Pacific Journal of Mathematics
975:Jacques Charles François Sturm
315:. Improved and popularized by
155:sums-of-squares of polynomials
1:
2442:162 (2005), no. 1, 195â234.
2408:245 (2003), no. 4, 725â760.
2032:10.1090/S0894-0347-05-00477-7
1586:237 (2001), no. 2, 259â273.
1254:Abh. Math. Sem. Univ. Hamburg
1182:Comessatti, Annibale (1914).
880:American Mathematical Society
454:sum of squares of polynomials
163:Krivine's Positivestellensatz
2256:97 (1989), no. 3, 467â483.
2150:10.1016/0040-9383(92)90025-d
2068:56, Fasc. 3 (1981), 339â351.
1924:Soviet Mathematics - Doklady
1893:Lecture Notes in Mathematics
1814:Proof of Gudkov's hypothesis
1573:42 (1993), no. 3, 969â984.
993:, vol. 52, pp. 39â51 (1856).
55:between them (in particular
1982:Mikhalkin, Grigory (2005).
1951:Tropical algebraic geometry
422:real quantifier elimination
313:real quantifier elimination
2526:
1911:Doklady Akademii Nauk SSSR
1519:Dubois, Donald W. (1967).
1051:Princeton University Press
850:Cambridge University Press
844:van den Dries, L. (1998).
411:PierceâBirkhoff conjecture
171:complex algebraic geometry
119:PierceâBirkhoff conjecture
2406:Mathematische Zeitschrift
1877:Marie-Françoise Coste-Roy
1679:10.1216/RMJ-1984-14-4-983
1584:Mathematische Zeitschrift
1405:Mathematische Zeitschrift
1306:10.1515/crll.1932.167.160
88:TarskiâSeidenberg theorem
2440:Inventiones Mathematicae
2355:Inventiones Mathematicae
2320:Inventiones Mathematicae
2254:Inventiones Mathematicae
1865:Inventiones Mathematicae
795:Real Algebraic Geometry.
57:real polynomial mappings
2510:Real algebraic geometry
2357:128 (2) (1997) 207â302.
1701:Krivine, J.-L. (1964).
229:Harnack's curve theorem
33:real algebraic geometry
2269:, 53, (1984), 143â151.
2267:Compositio Mathematica
2007:
1540:10.2140/pjm.1967.21.15
1053:, Princeton, NJ, 1965.
754:
702:
665:
588:
553:
499:1979 Michel Coste and
487:quantifier elimination
382:
353:
286:Hilbert's 17th problem
159:Hilbert's 17th problem
96:real analytic geometry
75:semialgebraic mappings
63:Semialgebraic geometry
2050:Annals of Mathematics
2008:
1770:Annals of Mathematics
1703:"Anneaux préordonnés"
1602:Annals of Mathematics
1456:10.1073/pnas.26.4.280
1389:Annals of Mathematics
1361:B. L. van der Waerden
1120:Mathematische Annalen
1007:Mathematische Annalen
929:Annals of Mathematics
755:
703:
701:{\displaystyle S^{n}}
666:
664:{\displaystyle S^{3}}
644:can be obtained from
589:
554:
552:{\displaystyle S^{n}}
383:
354:
298:B. L. van der Waerden
221:on real root counting
90:. Related fields are
35:is the sub-branch of
2052:113 (1981), 425â446.
1988:
1658:Mahé, Louis (1984).
1237:BĂ©la SzĆkefalvi-Nagy
911:Joseph B. J. Fourier
732:
685:
648:
563:
536:
363:
334:
151:positive polynomials
2370:, 36 (1997) 287â299
2287:K. SchmĂŒdgen, The
2211:Geometriae Dedicata
2094:53 (1981), 163â196.
1867:36, 295â312 (1976).
1794:Theodore S. Motzkin
1489:Kadison, Richard V.
1391:60 (1954), 365â374.
522:for curve counting.
512:Ragsdale conjecture
501:Marie-Françoise Roy
465:Gudkov's conjecture
319:in 1954. (Both use
265:Annibale Comessatti
204:Fourier's algorithm
179:convex optimization
167:commutative algebra
111:algebraic functions
49:algebraic equations
2224:10.1007/bf00147875
2180:, pp. 69â85,
2081:53 (1981), 79â162.
2003:
1861:Jean-Louis Verdier
1816:". V. A. Rokhlin.
1725:10.1007/BF02807438
1385:Abraham Seidenberg
1266:10.1007/BF02952512
1200:10.1007/BF02419577
1133:10.1007/BF01443605
1079:10.1007/PL00009443
1009:10 (1876), 189â199
750:
698:
661:
609:semialgebraic sets
605:semialgebraic sets
584:
549:
494:Jean-Louis Verdier
443:Whitney conditions
378:
349:
317:Abraham Seidenberg
302:semialgebraic sets
243:Hilbert's problems
147:real closed fields
67:semialgebraic sets
37:algebraic geometry
18:Real algebraic set
2322:107 (1992), 87â98
1960:978-3-7643-8309-1
1930:: 566â570. 1980.
1842:George E. Collins
1637:Birkhoff, Garrett
872:Khovanskii, A. G.
824:978-0-8218-4402-1
813:978-3-540-33098-1
520:tropical geometry
516:Grigory Mikhalkin
479:George E. Collins
103:Real plane curves
16:(Redirected from
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1300:(167): 160â196.
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1038:
1037:(1964), 275â280.
1025:
1019:
1016:
1010:
1003:C. G. A. Harnack
1000:
994:
984:
978:
972:
966:
963:Theodore Motzkin
960:
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461:Vladimir Rokhlin
450:Theodore Motzkin
432:Heisuke Hironaka
387:
385:
384:
379:
377:
376:
371:
358:
356:
355:
350:
348:
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342:
245:(especially the
212:Theodore Motzkin
187:valuation theory
181:, the theory of
92:o-minimal theory
65:is the study of
21:
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2040:
1991:
1986:
1985:
1981:
1980:
1976:
1961:
1948:
1947:
1943:
1921:
1918:(6): 1306â1309.
1904:
1903:
1899:
1887:
1883:
1875:
1871:
1859:
1855:
1840:
1836:
1830:Alberto Tognoli
1828:
1824:
1811:
1807:
1792:
1788:
1782:Hassler Whitney
1780:
1776:
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1615:10.2307/1969649
1595:
1594:
1590:
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1577:
1568:
1564:
1518:
1517:
1513:
1487:
1486:
1482:
1434:Stone, Marshall
1432:
1431:
1427:
1415:
1411:
1407:41 (1936), 1â17
1401:Herbert Seifert
1399:
1395:
1383:
1379:
1371:
1367:
1359:
1355:
1347:
1343:
1326:
1325:
1321:
1288:Krull, Wolfgang
1286:
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1281:
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1243:
1231:
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1219:
1215:
1181:
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1109:
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1104:
1062:
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1057:
1045:
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1026:
1022:
1017:
1013:
1001:
997:
987:Charles Hermite
985:
981:
973:
969:
961:
957:
942:10.2307/1967869
924:Dines, Lloyd L.
922:
921:
917:
909:
905:
890:
870:
869:
865:
843:
842:
838:
833:
826:; 0-8218-4402-4
815:; 3-540-33098-4
780:
735:
730:
729:
688:
683:
682:
651:
646:
645:
611:by Scheiderer.)
566:
561:
560:
539:
534:
533:
472:Alberto Tognoli
439:Hassler Whitney
397:Richard Kadison
366:
361:
360:
337:
332:
331:
328:Herbert Seifert
321:Sturm's theorem
284:'s solution of
219:Sturm's theorem
199:
183:quadratic forms
175:moment problems
145:(in particular
83:
23:
22:
15:
12:
11:
5:
2523:
2521:
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2512:
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2498:
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2479:External links
2477:
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2218:(3): 335â350.
2200:
2164:
2143:(2): 323â336.
2121:
2105:
2096:
2083:
2070:
2062:Selman Akbulut
2054:
2046:Selman Akbulut
2038:
2000:
1995:
1974:
1959:
1941:
1920:Translated in
1906:Viro, Oleg Ya.
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1609:(3): 405â421.
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1449:(4): 280â283.
1425:
1417:Selman Akbulut
1409:
1393:
1377:
1365:
1353:
1341:
1328:Baer, Reinhold
1319:
1279:
1241:
1225:
1213:
1194:(3): 215â283.
1174:
1155:Farkas, Julius
1146:
1127:(3): 342â350.
1111:Hilbert, David
1102:
1055:
1039:
1020:
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915:
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852:. p. 31.
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523:
518:applied it to
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115:Nash functions
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39:studying real
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1373:Alfred Tarski
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1088:2027.42/42421
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916:
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895:
891:
889:0-8218-4547-0
885:
881:
877:
873:
867:
864:
859:
855:
851:
847:
840:
837:
830:
825:
821:
817:
814:
810:
806:
804:
803:3-540-64663-9
800:
796:
792:
790:
789:0-387-97744-9
786:
782:
781:
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773:
769:
765:
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714:
710:
693:
689:
680:
677:
674:
656:
652:
643:
639:
636:
632:
628:
624:
622:denominators.
620:
617:
613:
610:
606:
602:
598:
595:
579:
576:
573:
544:
540:
531:
528:
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412:
408:
405:
401:
398:
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390:
373:
344:
329:
325:
322:
318:
314:
310:
309:Alfred Tarski
306:
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295:
292:
289:
287:
283:
279:
276:
275:Frigyes Riesz
272:
270:
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258:Farkas' lemma
255:
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233:Betti numbers
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143:ordered rings
140:
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93:
89:
80:
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72:
68:
64:
60:
58:
54:
50:
47:solutions to
46:
42:
38:
34:
30:
19:
2493:
2488:(PostScript)
2485:
2467:
2459:
2454:
2434:
2427:JĂĄnos KollĂĄr
2422:
2413:
2400:
2387:
2380:JĂĄnos KollĂĄr
2375:
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2327:
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2305:
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2209:
2203:
2187:math/0202086
2173:
2167:
2140:
2134:
2124:
2108:
2099:
2086:
2073:
2057:
2041:
2022:
2016:
1977:
1950:
1944:
1927:
1923:
1915:
1909:
1900:
1884:
1872:
1856:
1837:
1825:
1817:
1808:
1789:
1777:
1761:
1752:
1743:
1734:
1715:
1709:
1696:
1669:
1663:
1653:
1644:
1640:
1631:
1606:
1600:
1591:
1578:
1565:
1530:
1524:
1514:
1498:
1492:
1483:
1446:
1440:
1428:
1412:
1396:
1380:
1368:
1356:
1349:George PĂłlya
1344:
1335:
1331:
1322:
1297:
1291:
1282:
1257:
1253:
1244:
1228:
1216:
1191:
1187:
1177:
1168:
1162:
1149:
1124:
1118:
1105:
1070:
1064:
1058:
1042:
1034:
1023:
1014:
998:
982:
970:
958:
933:
927:
918:
906:
875:
866:
845:
839:
794:
771:
767:
726:JĂĄnos KollĂĄr
716:
713:JĂĄnos KollĂĄr
672:
641:
634:
630:
626:
268:
210:in 1919 and
191:model theory
135:Real algebra
134:
133:
124:
123:
100:
84:
71:inequalities
62:
61:
32:
26:
2025:: 313â377.
1718:: 307â326.
1250:Artin, Emil
1221:Lipót Fejér
1073:(1): 1â18.
1028:John Milnor
208:Lloyd Dines
81:Terminology
45:real-number
29:mathematics
2448:1082.14052
1969:1162.14300
1936:0422.14032
1597:Nash, John
898:0728.12002
876:Fewnomials
858:0953.03045
778:References
481:discovers
282:Emil Artin
101:Examples:
2240:117475206
1557:120262803
1533:: 15â19.
1501:: 39 pp,
1314:199547002
1274:122881707
1260:: 85â99.
1208:121297483
1141:177804714
1047:René Thom
615:subsets).
508:Oleg Viro
404:John Nash
107:polyhedra
2504:Category
2368:Topology
2333:Topology
2136:Topology
1647:: 41â69.
1475:16588355
1436:(1940).
1113:(1888).
874:(1991).
253:problem)
249:and the
214:in 1936.
53:mappings
2232:0765338
2196:2409689
2159:1167174
1849:0403962
1801:0223521
1688:0773148
1623:1969649
1549:0209200
1507:0044040
1466:1078172
1171:: 1â27.
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