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Book Information:
Authors:
I. M. Gelfand,
M. M. Kapranov and
A.V. Zelevinsky
Title:
Discriminants, resultants and multidimensional determinants
Additional book information:
Birkhäuser,
Boston,
1994,
vii + 523 pp.,
ISBN 0 817 63660 9,
$82.00$
Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409–448. MR 401754, DOI 10.2307/2318338
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
V. I. Arnol′d, Singularities of caustics and wave fronts, Mathematics and its Applications (Soviet Series), vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1151185, DOI 10.1007/978-94-011-3330-2
A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, Vol. IV, Math Sci Press, Brookline, Mass., 1975. MR 0457437
Gottfried Barthel, Friedrich Hirzebruch, and Thomas Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Friedr. Vieweg & Sohn, Braunschweig, 1987 (German). MR 912097, DOI 10.1007/978-3-322-92886-3
[Bez1] E. Bezout, Recherches sur le degre' des equations resultantes de l'evanouissement des inconnues et sur le moyen qu'il convient d' employer pour trouver les equations, Mem. Acad, Paris, 288-338 (1764).
[Bez2] E. Bezout, Theorie generale des equations algebriques, Paris, 471 pp. (1779).
[BM] E. Bierstone, P.D. Milman, Geometric and differential properties of subanalytic sets, Ann. Math., II. Ser. 147, No.3, 731-785 (1998). CMP 98:16
[BrN]A. Brill, M. Noether, Die Entwicklung der Theorie der algebraische Funktionen in älterer und neuerer Zeit, Jahresber. der Deutsche Math. Verein. t. III (1892-93), 111-566.
Sylvain E. Cappell and Julius L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 62–69. MR 1217352, DOI 10.1090/S0273-0979-1994-00436-7
F. Catanese, On the rationality of certain moduli spaces related to curves of genus $4$, Algebraic geometry (Ann Arbor, Mich., 1981) Lecture Notes in Math., vol. 1008, Springer, Berlin, 1983, pp. 30–50. MR 723706, DOI 10.1007/BFb0065697
[Cr] G. Cramer, Introduction a l' analyse des lignes courbes algebriques, Geneve, 680 pp. (1750).
Julian Lowell Coolidge, A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959. MR 0120551
V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177 (French). MR 902592, DOI 10.1090/conm/067/902592
[Di] U. Dini, Lezioni di Analisi infinitesimale, Vol. I(1-2), II. Nistri Lischi, Pisa, 1907, 1909, CI + 720, XIII+468 pp.
Gerd Faltings, The determinant of cohomology in etale topology, Arithmetic geometry (Cortona, 1994) Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 157–168. MR 1472496
[Fl] H. Flenner, Emmy Noether and the development of commutative algebra in `The Heritage of Emmy Noether', Israel Math. Conf. Proc. 12 (1999), 23-38. CMP 99:07
[Fr] W. Franz, Über die Torsion einer Ueberdeckung, J. Reine Angew. Math. 173, 245-254 (1935).
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
[GoK1] P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie I: Determinanten, Teubner, Leipzig, 1885.
[GoK2] P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie II: Binäre Formen, Teubner, Leipzig, 1887.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
S. Hosono, B. H. Lian, and S.-T. Yau, GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Comm. Math. Phys. 182 (1996), no. 3, 535–577. MR 1461942
G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. MR 437541, DOI 10.7146/math.scand.a-11642
John Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147. MR 141115, DOI 10.2307/1970268
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
[Moe] A.F. Möbius, Der barycentrische Calcul, Barth Verlag, Leipzig, 1827, 480 pp.
David Mumford, Algebraic geometry. I, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Complex projective varieties; Reprint of the 1976 edition. MR 1344216
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
[Po] J.V. Poncelet, Traite' des Proprietes Projectives des Figures, Tomes I et II, Gauthier-Villars, Paris, 1822, deux. Ed. 1865, XXXII, 428 and 452 pp.
[Pr] C. Procesi, 150 years of invariant theory in `The Heritage of Emmy Noether', Israel Math. Conf. Proc. 12 (1999), 5-21. CMP 99:07
D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI 10.2307/1970909
[Rei1] K. Reidemeister, Überdeckungen von Komplexen, J. Reine Angew. Math. 173, 164-173 (1935).
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
[Ro] T.G. Room, The geometry of determinantal loci, Cambridge: Univ. Press. XXVIII, 483 pp. (1938).
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
E. I. Shustin, Surjectivity of the resultant map: a solution to the inverse Bezout problem, Comm. Algebra 23 (1995), no. 3, 1145–1163. MR 1316754, DOI 10.1080/00927879508825271
Andrew J. Sommese and Charles W. Wampler, Numerical algebraic geometry, The mathematics of numerical analysis (Park City, UT, 1995) Lectures in Appl. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 749–763. MR 1421365
Bernd Sturmfels, Sparse elimination theory, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 264–298. MR 1253995
[vdW] B.L. Van der Waerden, Einführung in die Algebraische Geometrie, Grundlehren d. math. Wiss. in Einzeldarstell. 51, Berlin: Julius Springer. VII, 247 pp. (1939).
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Andrew H. Wallace, Homology theory on algebraic varieties, International Series of Monographs on Pure and Applied Mathematics, Vol. 6, Pergamon Press, New York-London-Paris-Los Angeles, 1958. MR 0093522
[We] H. Weber, Lehrbuch der Algebra, I, II, Vieweg, Braunschweig, 1895-1896.
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Auguste Dick, Emmy Noether, 1882–1935, Birkhäuser, Boston, Mass., 1981. Translated from the German by Heidi I. Blocher; With contributions by B. L. van der Waerden, Hermann Weyl and P. S. Alexandrov [P. S. Aleksandrov]. MR 610977
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- [Abh]
- Abhyankar, Shreeram S., Historical ramblings in algebraic geometry and related algebra, Am. Math. Mon. 83, 409-448 (1976). MR 53:5581
- [ACGH]
- E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, Vol. I, Grundlehren der mathematischen Wissenschaften, 267. New York, Springer-Verlag. XVI, 386 p. (1985). MR 86h:14019
- [Ar]
- V.I. Arnold, Singularities of caustics and wave fronts, Transl. from the Russian, Mathematics and Its Applications (Soviet Series), 62. Dordrecht: Kluwer Academic Publishers. xiii, (1990), 259 pp. MR 93b:58019
- [AMRT]
- A. Ash, D. Mumford, M. Rapoport, Y.Tai, Smooth compactification of locally symmetric varieties, Math Sci Press. IV, Brookline, Mass. (1975), 335 pp. MR 56:15642
- [BHH]
- G. Barthel, F. Hirzebruch, T. Höfer, Geradenkonfigurationen und algebraische Flächen Vieweg Aspekte der Mathematik, Bd. 4. (1987). MR 89a:14045
- [Bez1]
- E. Bezout, Recherches sur le degre' des equations resultantes de l'evanouissement des inconnues et sur le moyen qu'il convient d' employer pour trouver les equations, Mem. Acad, Paris, 288-338 (1764).
- [Bez2]
- E. Bezout, Theorie generale des equations algebriques, Paris, 471 pp. (1779).
- [BM]
- E. Bierstone, P.D. Milman, Geometric and differential properties of subanalytic sets, Ann. Math., II. Ser. 147, No.3, 731-785 (1998). CMP 98:16
- [BrN]
- A. Brill, M. Noether, Die Entwicklung der Theorie der algebraische Funktionen in älterer und neuerer Zeit, Jahresber. der Deutsche Math. Verein. t. III (1892-93), 111-566.
- [CSh]
- S.E. Cappell, J.L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Am. Math. Soc., New Ser. 30, No. 1, 62-69 (1994). MR 94f:14018
- [Cat]
- F. Catanese, On the rationality of certain moduli spaces related to curves of genus 4, Algebraic geometry, Proc. Conf., Ann Arbor, Mich., 1981, Lect. Notes Math. 1008, 28-50 (1983). MR 85b:14032
- [Cr]
- G. Cramer, Introduction a l' analyse des lignes courbes algebriques, Geneve, 680 pp. (1750).
- [Coo]
- J.L. Coolidge, A treatise on algebraic plane curves, Oxford Univ. Press (1931), reprinted by Dover Publications, New York, Inc. XXIV, (1959), 513 pp. MR 22:11302
- [D]
- V.I. Danilov, Geometry of toric varieties, Russ. Math. Surv. 33, No.2, 85-134 (1978). MR 80g:14001
- [De]
- P. Deligne, Le determinant de la cohomologie, Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata, Calif., 1985, Contemp. Math. 67, 93-117 (1987). MR 89b:32038
- [Di]
- U. Dini, Lezioni di Analisi infinitesimale, Vol. I(1-2), II. Nistri Lischi, Pisa, 1907, 1909, CI + 720, XIII+468 pp.
- [Fa]
- G. Faltings, The determinant of cohomology in etale topology, Arithmetic geometry, Proceedings, Cortona, 1994, Cambridge University Press, Symp. Math. 37, 157-168 (1997). MR 98k:14023
- [Fl]
- H. Flenner, Emmy Noether and the development of commutative algebra in `The Heritage of Emmy Noether', Israel Math. Conf. Proc. 12 (1999), 23-38. CMP 99:07
- [Fr]
- W. Franz, Über die Torsion einer Ueberdeckung, J. Reine Angew. Math. 173, 245-254 (1935).
- [Fu1]
- W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 2. Berlin Springer-Verlag. XI, (1984), 470 pp. MR 85k:14004
- [Fu2]
- W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies. 131. Princeton University Press (1993), 157 pp. MR 94g:14028
- [GKZ]
- I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston (1994), vii + 523 pp. MR 95e:14045
- [GoK1]
- P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie I: Determinanten, Teubner, Leipzig, 1885.
- [GoK2]
- P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie II: Binäre Formen, Teubner, Leipzig, 1887.
- [Hart]
- R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics. 52. Springer-Verlag. (1977), XVI+496 pp. MR 57:3116
- [HLY]
- S. Hosono, B.H. Lian, S.-T. Yau, GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Commun. Math. Phys. 182, No.3, 535-577 (1996). MR 98g:14042
- [KKMS]
- G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I. Lecture Notes in Mathematics, 339, Berlin-Heidelberg-New York: Springer-Verlag (1973), 209 pp. MR 49:299
- [Knud-Mum]
- F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on ``det'' and ``Div'', Math. Scandinav. 39, 19-55 (1976). MR 55:10465
- [Miln1]
- J. W. Milnor, A duality theorem for Reidemeister torsion, Ann. Math., II. Ser. 76, 137-147 (1962). MR 25:4526
- [Miln2]
- J. W. Milnor, Whitehead torsion, Bull. Am. Math. Soc. 72, 358-426 (1966). MR 33:4922
- [Moe]
- A.F. Möbius, Der barycentrische Calcul, Barth Verlag, Leipzig, 1827, 480 pp.
- [Mum]
- D. Mumford, Algebraic geometry. I: Complex projective varieties, Grundlehren der math. Wissenschaften. 221. Berlin -Heidelberg - New York: Springer-Verlag. X, 186 pp. (1976). MR 96d:14001
- [O]
- T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik 3. F. B. 15. Springer-Verlag (1988) 212 pp. MR 88m:14038
- [Po]
- J.V. Poncelet, Traite' des Proprietes Projectives des Figures, Tomes I et II, Gauthier-Villars, Paris, 1822, deux. Ed. 1865, XXXII, 428 and 452 pp.
- [Pr]
- C. Procesi, 150 years of invariant theory in `The Heritage of Emmy Noether', Israel Math. Conf. Proc. 12 (1999), 5-21. CMP 99:07
- [Q]
- D. Quillen, Determinants of Cauchy-Riemann operators on a Riemann surface, Funct. Anal. Appl. 19, 31-34 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 37-41 (1985). MR 86g:32035
- [RS1]
- D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7, 145-210 (1971). MR 45:4447
- [RS2]
- D.B. Ray, I.M. Singer, Analytic torsion for complex manifolds, Ann. of Math., II. Ser. 98, 154-177 (1973). MR 52:4344
- [Rei1]
- K. Reidemeister, Überdeckungen von Komplexen, J. Reine Angew. Math. 173, 164-173 (1935).
- [Rei2]
- K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48, 226-239 (1939). MR 1:105h
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- T.G. Room, The geometry of determinantal loci, Cambridge: Univ. Press. XXVIII, 483 pp. (1938).
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- E.I. Shustin, Surjectivity of the resultant map : a solution to the inverse Bezout problem, Comm. in Alg., 23 (3), 1145-1163 (1995). MR 96a:12004
- [SoWa]
- A.J. Sommese, C.W. Wampler, Numerical algebraic geometry in `The mathematics of numerical analysis', 1995 AMS-SIAM Park City summer seminar, American Mathematical Society. Lect. Appl. Math. 32, 749-763 (1996). MR 98d:14079
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- B. Sturmfels, Sparse elimination theory in `Computational algebraic geometry and commutative algebra' Proceedings, Cortona, 1991. Cambridge University Press. Symp. Math. 34, 264-298 (1993). MR 94k:13035
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- B.L. Van der Waerden, Einführung in die Algebraische Geometrie, Grundlehren d. math. Wiss. in Einzeldarstell. 51, Berlin: Julius Springer. VII, 247 pp. (1939).
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- A.H. Wallace, Tangency and duality over arbitrary fields, Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956). MR 18:234b
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- A.H. Wallace, Homology theory on algebraic varieties, Pergamon Press, London, New York, Paris, Los Angeles (1958), VIII, 114 pp. MR 20:46
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- A. Weil, Foundations of algebraic geometry, American Mathematical Society Colloquium Publications. 29, New York: American Mathematical Society. XIX, 299 p. (1946). MR 9:303c
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- H. Weyl, Symmetry, Princeton: Princeton University Press. (1952), 168 pp. MR 14:16a
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Review Information:
Reviewer:
Fabrizio Catanese
Affiliation:
Georg-August-Universität Göttingen
Email:
catanese@uni-math.gwdg.de
Journal:
Bull. Amer. Math. Soc.
37 (2000), 183-198
DOI:
https://doi.org/10.1090/S0273-0979-99-00858-7
Published electronically:
December 21, 1999
Review copyright:
© Copyright 2000
American Mathematical Society