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

Yu, Jie Tai

doi:10.1090/S0002-9939-1995-1216829-4

A product formula for minimal polynomials and degree bounds for inverses of polynomial automorphisms
HTML articles powered by AMS MathViewer

by Jie Tai Yu PDF

Proc. Amer. Math. Soc. 123 (1995), 343-349 Request permission

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

Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
Hyman Bass, Edwin H. Connell, and David Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330. MR 663785, DOI 10.1090/S0273-0979-1982-15032-7
Wei Li and Jie Tai Yu, Computing minimal polynomials and the degree of unfaithfulness, Comm. Algebra 21 (1993), no. 10, 3557–3569. MR 1231617, DOI 10.1080/00927879308824749
Wei Li and Jie Tai Yu, Reconstructing birational maps from their face functions, Manuscripta Math. 76 (1992), no. 3-4, 353–366. MR 1185025, DOI 10.1007/BF02567766
James H. McKay and Stuart Sui Sheng Wang, An inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 40 (1986), no. 3, 245–257. MR 836651, DOI 10.1016/0022-4049(86)90044-7
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732

Product formulas for sparse resultants

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
Bernd Sturmfels and Jie Tai Yu, Minimal polynomials and sparse resultants, Zero-dimensional schemes (Ravello, 1992) de Gruyter, Berlin, 1994, pp. 317–324. MR 1292495
Stuart Sui Sheng Wang, A Jacobian criterion for separability, J. Algebra 65 (1980), no. 2, 453–494. MR 585736, DOI 10.1016/0021-8693(80)90233-1
Jie Tai Yu, Face polynomials and inversion formula, J. Pure Appl. Algebra 78 (1992), no. 2, 213–219. MR 1161345, DOI 10.1016/0022-4049(92)90099-2
Jie Tai Yu, Computing minimal polynomials and the inverse via GCP, Comm. Algebra 21 (1993), no. 7, 2279–2294. MR 1218498, DOI 10.1080/00927879308824677