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

DOI:
https://doi.org/10.1090/S0002-9939-1995-1216829-4

MathSciNet review:
1216829

Full-text PDF Free Access

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.

- Shreeram S. Abhyankar,
*Algebraic geometry for scientists and engineers*, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR**1075991** - 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-4049%2886%2990044-7 - David Mumford,
*Algebraic geometry. I*, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR**0453732**
P. Pederson and B. Sturmfels, - 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 https://doi.org/10.1016/0021-8693%2880%2990233-1 - Jie Tai Yu,
*Face polynomials and inversion formula*, J. Pure Appl. Algebra**78**(1992), no. 2, 213–219. MR**1161345**, DOI https://doi.org/10.1016/0022-4049%2892%2990099-2 - Jie Tai Yu,
*Computing minimal polynomials and the inverse via GCP*, Comm. Algebra**21**(1993), no. 7, 2279–2294. MR**1218498**, DOI https://doi.org/10.1080/00927879308824677

*Product formulas for sparse resultants*, J. Algebra (to appear) (1993).

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