Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A chain rule for multivariable resultants


Authors: Charles Ching-an Cheng, James H. McKay and Stuart Sui Sheng Wang
Journal: Proc. Amer. Math. Soc. 123 (1995), 1037-1047
MSC: Primary 12D10; Secondary 13B25
MathSciNet review: 1227515
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a chain rule for the multivariable resultant, which is similar to the familiar chain rule for the Jacobian matrix. Specifically, given two homogeneous polynomial maps $ {K^n} \to {K^n}$ for a commutative ring K, such that their composition is a homogeneous polynomial map, the resultant of the composition is the product of appropriate powers of resultants of the individual maps.


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

  • [1] N. Bourbaki, Commutative algebra, Elements of Mathematics, Addison-Wesley, Reading, MA, 1972.
  • [2] F. Faà de Bruno, Théorie des formes binaires, Résumé des leçons faites à l'Université de Turin, Librairie Brero succr. de P. Marietti, Turin, 1876.
  • [3] M. Chardin, Contributions à l'algèbre commutative effective et à la théorie de l'élimination, Thèse de l'Université Pierre et Marie Curie (Paris VI), 1990, preprint of Centre de Mathématiques de l'Ecole Polytechnique F-91128 Palaiseau France.
  • [4] C. C. Cheng, J. H. McKay, and S. S.-S. Wang, Chain rule for multivariable resultants, Abstracts Amer. Math. Soc. 13 (1992), 242 (873-13-109).
  • [5] Arno van den Essen and Michał Kwieciński, On the reconstruction of polynomial automorphisms from their face polynomials, J. Pure Appl. Algebra 80 (1992), no. 3, 327–336. MR 1170718, 10.1016/0022-4049(92)90150-E
  • [6] I. M. Gel′fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Discriminants of polynomials in several variables, Funktsional. Anal. i Prilozhen. 24 (1990), no. 1, 1–4 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 1, 1–4. MR 1052262, 10.1007/BF01077912
  • [7] N. Kravitsky and Z. Waksman, On some resultant identities, Linear Algebra Appl. 122/123/124 (1989), 3–21. MR 1019980, 10.1016/0024-3795(89)90645-9
  • [8] Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the German by Michael Ackerman; With a preface by David Mumford. MR 789602
  • [9] F. S. Macaulay, Some formulæin elimination, Proc. London Math. Soc. 35 (1903), 3-27.
  • [10] -, The algebraic theory of modular systems, Cambridge Tracts in Math., vol. 19, Cambridge Univ. Press, Cambridge, 1916.
  • [11] J. H. McKay and S. S.-S. Wang, Chain rule for resultant, Abstracts Amer. Math. Soc. 8 (1987), 329 (836-13-122).
  • [12] James H. McKay and Stuart Sui Sheng Wang, A chain rule for the resultant of two polynomials, Arch. Math. (Basel) 53 (1989), no. 4, 347–351. MR 1015998, 10.1007/BF01195214
  • [13] James H. McKay and Stuart Sui Sheng Wang, A chain rule for the resultant of two homogeneous polynomials, Arch. Math. (Basel) 56 (1991), no. 4, 352–361. MR 1094422, 10.1007/BF01198221
  • [14] Patrice Philippon, Critères pour l’indépendance algébrique, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 5–52 (French). MR 876159

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12D10, 13B25

Retrieve articles in all journals with MSC: 12D10, 13B25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1227515-9
Keywords: Multivariable resultant, common zeros, generic polynomials, Jacobian Conjecture, chain rule, Nullstellensatz, discriminant, isobaric property, invariant
Article copyright: © Copyright 1995 American Mathematical Society