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 product formula for minimal polynomials and degree bounds for inverses of polynomial automorphisms

Author: Jie Tai Yu
Journal: Proc. Amer. Math. Soc. 123 (1995), 343-349
MSC: Primary 12E05; Secondary 12F05, 12Y05
MathSciNet review: 1216829
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By means of Galois theory, we give a product formula for the minimal polynomial G of $ \{ {f_0},{f_1}, \ldots ,{f_n}\} \subset K[{x_1}, \ldots ,{x_n}]$ which contains n algebraically independent elements, where K is a field of characteristic zero. As an application of the product formula, we give a simple proof of Gabber's degree bound inequality for the inverse of a polynomial automorphism.

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

  • [1] S. S. Abhyankar, Algebraic geometry for scientists and engineers, Amer. Math. Soc., Providence, RI, 1990. MR 1075991 (92a:14001)
  • [2] H. Bass, E. Connell, and D. Wright, The Jacobian conjecture: reduction on degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 287-330. MR 663785 (83k:14028)
  • [3] W. Li and J.-T. Yu, Computing minimal polynomials and the degree of unfaithfulness, Comm. Algebra 21 (1993), 3557-3569. MR 1231617 (94h:13020)
  • [4] -, Reconstructing birational maps from their face functions, Manuscripta Math. 76 (1992), 353-366. MR 1185025 (93j:14014)
  • [5] J. MaKay and S. S.-S. Wang, An inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 40 (1986), 245-257. MR 836651 (87j:12003)
  • [6] D. Mumford, Algebraic geometry I. Complex projective varieties, Springer-Verlag, New York, 1976. MR 0453732 (56:11992)
  • [7] P. Pederson and B. Sturmfels, Product formulas for sparse resultants, J. Algebra (to appear) (1993).
  • [8] B. Sturmfels, Sparse elimination theory, Proceedings of Computational Algebraic Geometry and Commutative Algebra, Cortona, Italy, 1992. MR 1253995 (94k:13035)
  • [9] B. Sturmfels and J.-T. Yu, Minimal polynomials and sparse resultants, Proceedings of the Zero Dimensional Conference (Ravello, Italy, June 8-13, 1992), Cortona, Italy, 1993. MR 1292495 (95g:12001)
  • [10] S. S.-S. Wang, A Jacobian criterion for separability, J. Algebra 65 (1980), 453-494. MR 585736 (83e:14010)
  • [11] J.-T. Yu, Face polynomials and inversion formula, J. Pure Appl. Algebra 78 (1992), 213-219. MR 1161345 (93b:13010)
  • [12] -, Computing minimal polynomials and the inverse via GCP, Comm. Algebra 21 (1993), 2279-2294. MR 1218498 (94d:12002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12E05, 12F05, 12Y05

Retrieve articles in all journals with MSC: 12E05, 12F05, 12Y05

Additional Information

Keywords: Minimal polynomials, Galois theory, product formula, polynomial automorphisms
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society