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
DOI: https://doi.org/10.1090/S0002-9939-1995-1227515-9
MathSciNet review: 1227515
Full-text PDF

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] A. van den Essen and M. Kwieciński, On the reconstruction of polynomial automorphisms from their face polynomials, J. Pure Appl. Algebra 80 (1992), 327-336. MR 1170718 (93j:14015)
  • [6] I. M. Gel'fand, A. V. Zelevinskii, and M. M. Kapranov, Discriminants of polynomials in many variables, Funct. Anal. Appl. 24 (1990), 1-4. MR 1052262 (91e:14047)
  • [7] N. Kravitsky and Z. Waksman, On some resultant identities, Linear Algebra Appl. 122-124 (1989), 3-21. MR 1019980 (90k:15005)
  • [8] E. Kunz, Introduction to commutative algebra and algebraic geometry, Einführung in die kommutative Algebra und algebraische Geometrie, Birkhäuser, Boston, MA, 1985. MR 789602 (86e:14001)
  • [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] -, A chain rule for the resultant of two polynomials, Arch. Math. 53 (1989), 347-351. MR 1015998 (90h:12006)
  • [13] -, A chain rule for the resultant of two homogeneous polynomials, Arch. Math. 56 (1991), 352-361. MR 1094422 (92a:12006)
  • [14] P. Philippon, Critères pour l'indépendance algébrique, Inst. Hautes Études Sci. Publ. Mat. 64 (1986), 5-52. MR 876159 (88h:11048)

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: https://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

American Mathematical Society