Book Review
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3090426
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Book Information:
Author:
Frank Sottile
Title:
Real solutions to equations from geometry
Additional book information:
University Lecture Series, Vol. 57,
American Mathematical Society,
Providence, Rhode Island,
2011,
x+200 pp.,
ISBN 978-0-8218-5331-3,
List Price: US$47, Member Price US$37.60
Selman Akbulut, Real algebraic structures, Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, 2006, pp. 49–58. MR 2282009
S. Akbulut and H. King, Submanifolds and homology of nonsingular real algebraic varieties, Amer. J. Math. 107 (1985), no. 1, 45–83. MR 778089, DOI 10.2307/2374457
Selman Akbulut and Henry C. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. (2) 113 (1981), no. 3, 425–446. MR 621011, DOI 10.2307/2006992
S. Akbulut and H. King, All knots are algebraic, Comment. Math. Helv. 56 (1981), no. 3, 339–351. MR 639356, DOI 10.1007/BF02566217
V. I. Arnol′d, The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 1–9 (Russian). MR 0286790
E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.
S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22 (1999), no. 1, 1–18. MR 1692627, DOI 10.1007/PL00009443
J. Dieudonné, The historical development of algebraic geometry, Amer. Math. Monthly 79 (1972), 827–866. MR 308117, DOI 10.2307/2317664
A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2) 155 (2002), no. 1, 105–129. MR 1888795, DOI 10.2307/3062151
A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 443–472. MR 2352567, DOI 10.1112/plms/pdm012
D. A. Gudkov and G. A. Utkin, The topology of curves of degree and surfaces of degree, Uchen. Zap. Gorkov. Univ., vol. 87, 1969 (Russian), English transl., Transl. AMS 112.
Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189–198 (German). MR 1509883, DOI 10.1007/BF01442458
David Hilbert, Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), no. 3, 342–350 (German). MR 1510517, DOI 10.1007/BF01443605
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Ilia Itenberg, Contre-examples à la conjecture de Ragsdale, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 3, 277–282 (French, with English and French summaries). MR 1233426
V. M. Harlamov, New congruences for the Euler characteristic of real algebraic manifolds, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 74–78 (Russian). MR 0331403
Morris Kline, Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. MR 0472307
János Kollár, The Nash conjecture for threefolds, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 63–73. MR 1641168, DOI 10.1090/S1079-6762-98-00049-3
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 173265
G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology 36 (1997), no. 1, 287–299. MR 1410476, DOI 10.1016/0040-9383(95)00062-3
J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 161339, DOI 10.1090/S0002-9939-1964-0161339-9
T. S. Motzkin, The arithmetic-geometric inequality, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) Academic Press, New York, 1967, pp. 205–224. MR 0223521
Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), no. 2, 863–881. MR 2552110, DOI 10.4007/annals.2009.170.863
John Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405–421. MR 50928, DOI 10.2307/1969649
I. G. Petrovskiĭ and O. A. Oleĭnik, On the topology of real algebraic surfaces, Izvestiya Akad. Nauk SSSR. Ser. Mat. 13 (1949), 389–402 (Russian). MR 0034600
G. Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928) 141–145, in: R. P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309–313.
V. Ragsdale, On the Arrangement of the Real Branches of Plane Algebraic Curves, Amer. J. Math. 28 (1906), no. 4, 377–404. MR 1505999, DOI 10.2307/2370070
J. C. F. Sturm, Mémoire sur la résolution des équations numériques, Mémoires divers présenteés par des savant étrangers à l’Académie Royale des Sciences de l’Institut de France, sciences mathématiques et physiques, 6 (1835) 273–318.
A. Tarski, 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) 206–207; and in Fund. Math. 17 (1931) 210–239.
René Thom, Sur l’homologie des variétés algébriques réelles, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 255–265 (French). MR 0200942
A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 167–185. MR 396571
O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187–200. MR 770238, DOI 10.1007/BFb0099934
O. Ja. Viro, Curves of degree $7$, curves of degree $8$ and the Ragsdale conjecture, Dokl. Akad. Nauk SSSR 254 (1980), no. 6, 1306–1310 (Russian). MR 592496
O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187–200. MR 770238, DOI 10.1007/BFb0099934
O. Ya. Viro, Real plane algebraic curves: constructions with controlled topology, Algebra i Analiz 1 (1989), no. 5, 1–73 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1059–1134. MR 1036837
Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, DOI 10.1007/s00222-005-0445-0
References
- Selman Akbulut, Real algebraic structures, Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, 2006, pp. 49–58. MR 2282009 (2007j:14087)
- S. Akbulut and H. King, Submanifolds and homology of nonsingular real algebraic varieties, Amer. J. Math. 107 (1985), no. 1, 45–83. MR 778089 (86m:14017), DOI 10.2307/2374457
- Selman Akbulut and Henry C. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. (2) 113 (1981), no. 3, 425–446. MR 621011 (83b:58003), DOI 10.2307/2006992
- S. Akbulut and H. King, All knots are algebraic, Comment. Math. Helv. 56 (1981), no. 3, 339–351. MR 639356 (83m:57005), DOI 10.1007/BF02566217
- V. I. Arnol′d, The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 1–9 (Russian). MR 0286790 (44 \#3999)
- E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.
- S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22 (1999), no. 1, 1–18. MR 1692627 (2000d:14061), DOI 10.1007/PL00009443
- J. Dieudonné, The historical development of algebraic geometry, Amer. Math. Monthly 79 (1972), 827–866. MR 0308117 (46 \#7232)
- A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2) 155 (2002), no. 1, 105–129. MR 1888795 (2003c:58028), DOI 10.2307/3062151
- A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 443–472. MR 2352567 (2008g:31008), DOI 10.1112/plms/pdm012
- D. A. Gudkov and G. A. Utkin, The topology of curves of degree and surfaces of degree, Uchen. Zap. Gorkov. Univ., vol. 87, 1969 (Russian), English transl., Transl. AMS 112.
- Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189–198 (German). MR 1509883, DOI 10.1007/BF01442458
- David Hilbert, Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), no. 3, 342–350 (German). MR 1510517, DOI 10.1007/BF01443605
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184 (33 \#7333)
- Ilia Itenberg, Contre-examples à la conjecture de Ragsdale, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 3, 277–282 (French, with English and French summaries). MR 1233426 (94f:14050)
- V. M. Kharlamov, New congruences for the Euler characteristic of real algebraic manifolds, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 74–78 (Russian). MR 0331403 (48 \#9736)
- Morris Kline, Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. MR 0472307 (57 \#12010)
- János Kollár, The Nash conjecture for threefolds, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 63–73 (electronic). MR 1641168 (99k:57038), DOI 10.1090/S1079-6762-98-00049-3
- S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449–474. MR 0173265 (30 \#3478)
- G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology 36 (1997), no. 1, 287–299. MR 1410476 (98f:57046), DOI 10.1016/0040-9383(95)00062-3
- J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 0161339 (28 \#4547)
- T. S. Motzkin, The arithmetic-geometric inequality, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967, pp. 205–224. MR 0223521 (36 \#6569)
- Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), no. 2, 863–881. MR 2552110 (2011b:17065), DOI 10.4007/annals.2009.170.863
- John Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405–421. MR 0050928 (14,403b)
- I. G. Petrovskiĭ and O. A. Oleĭnik, On the topology of real algebraic surfaces, Izvestiya Akad. Nauk SSSR. Ser. Mat. 13 (1949), 389–402 (Russian). MR 0034600 (11,613h)
- G. Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928) 141–145, in: R. P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309–313.
- V. Ragsdale, On the arrangement of the real branches of plane algebraic curves, Amer. J. Math. 28 (1906), no. 4, 377–404. MR 1505999, DOI 10.2307/2370070
- J. C. F. Sturm, Mémoire sur la résolution des équations numériques, Mémoires divers présenteés par des savant étrangers à l’Académie Royale des Sciences de l’Institut de France, sciences mathématiques et physiques, 6 (1835) 273–318.
- A. Tarski, 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) 206–207; and in Fund. Math. 17 (1931) 210–239.
- René Thom, Sur l’homologie des variétés algébriques réelles, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 255–265 (French). MR 0200942 (34 \#828)
- A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 167–185. MR 0396571 (53 \#434)
- O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187–200. MR 770238 (87i:14029), DOI 10.1007/BFb0099934
- O. Ja. Viro, Curves of degree $7$, curves of degree $8$ and the Ragsdale conjecture, Dokl. Akad. Nauk SSSR 254 (1980), no. 6, 1306–1310 (Russian). MR 592496 (82e:14067)
- O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187–200. MR 770238 (87i:14029), DOI 10.1007/BFb0099934
- O. Ya. Viro, Real plane algebraic curves: constructions with controlled topology, Algebra i Analiz 1 (1989), no. 5, 1–73 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1059–1134. MR 1036837 (91b:14078)
- Jean-Yves Welschinger, Invariants of real symplectic $4$-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329 (2006i:53123), DOI 10.1007/s00222-005-0445-0
Review Information:
Reviewer:
Michael Shapiro
Affiliation:
Michigan State University
Journal:
Bull. Amer. Math. Soc.
50 (2013), 675-680
DOI:
https://doi.org/10.1090/S0273-0979-2013-01397-1
Published electronically:
January 17, 2013
Review copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
