Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

Full text of review: PDF   This review is available free of charge.
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, $82.00$

References [Enhancements On Off] (What's this?)

  • Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409–448. MR 401754, DOI
  • 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
  • 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
  • A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975. Lie Groups: History, Frontiers and Applications, Vol. IV. 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
  • 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). E. Bezout, Theorie generale des equations algebriques, Paris, 471 pp. (1779). 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 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
  • 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
  • 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
  • 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
  • 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 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
  • 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
  • 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
  • P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie I: Determinanten, Teubner, Leipzig, 1885. P. Gordan, G. Kerschensteiner, Vorlesungen über Invariantentheorie II: Binäre Formen, Teubner, Leipzig, 1887.
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. 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
  • John Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147. MR 141115, DOI
  • J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI
  • 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
  • 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. 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
  • D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI
  • K. Reidemeister, Überdeckungen von Komplexen, J. Reine Angew. Math. 173, 164-173 (1935). K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48, 226-239 (1939). MR 1:105h T.G. Room, The geometry of determinantal loci, Cambridge: Univ. Press. XXVIII, 483 pp. (1938). G. Sansone, R. Conti, Equazioni differenziali non lineari, Consiglio Naz. Ricerche, Monografie Mat. 3. Roma: Edizioni Cremonese XIX, 647 p. (1956). MR 19:547c
  • 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
  • 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
  • B.L. Van der Waerden, Einführung in die Algebraische Geometrie, Grundlehren d. math. Wiss. in Einzeldarstell. 51, Berlin: Julius Springer. VII, 247 pp. (1939). A.H. Wallace, Tangency and duality over arbitrary fields, Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956). MR 18:234b
  • 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
  • H. Weber, Lehrbuch der Algebra, I, II, Vieweg, Braunschweig, 1895-1896. A. Weil, Foundations of algebraic geometry, American Mathematical Society Colloquium Publications. 29, New York: American Mathematical Society. XIX, 299 p. (1946). MR 9:303c H. Weyl, Symmetry, Princeton: Princeton University Press. (1952), 168 pp. MR 14:16a
  • 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
  • J.H.C. Whitehead, Simple homotopy type, Am. J. Math. 72, 1-57 (1950). MR 11:735c

Review Information:

Reviewer: Fabrizio Catanese
Affiliation: Georg-August-Universität Göttingen
Journal: Bull. Amer. Math. Soc. 37 (2000), 183-198
Published electronically: December 21, 1999
Review copyright: © Copyright 2000 American Mathematical Society