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

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Abstract | References | Similar Articles | Additional Information

Abstract: By means of Galois theory, we give a product formula for the minimal polynomial *G* of 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.

**[1]**Shreeram S. Abhyankar,*Algebraic geometry for scientists and engineers*, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR**1075991****[2]**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**, https://doi.org/10.1090/S0273-0979-1982-15032-7**[3]**Wei Li and Jie Tai Yu,*Computing minimal polynomials and the degree of unfaithfulness*, Comm. Algebra**21**(1993), no. 10, 3557–3569. MR**1231617**, https://doi.org/10.1080/00927879308824749**[4]**Wei Li and Jie Tai Yu,*Reconstructing birational maps from their face functions*, Manuscripta Math.**76**(1992), no. 3-4, 353–366. MR**1185025**, https://doi.org/10.1007/BF02567766**[5]**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**, https://doi.org/10.1016/0022-4049(86)90044-7**[6]**David Mumford,*Algebraic geometry. I*, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR**0453732****[7]**P. Pederson and B. Sturmfels,*Product formulas for sparse resultants*, J. Algebra (to appear) (1993).**[8]**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****[9]**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****[10]**Stuart Sui Sheng Wang,*A Jacobian criterion for separability*, J. Algebra**65**(1980), no. 2, 453–494. MR**585736**, https://doi.org/10.1016/0021-8693(80)90233-1**[11]**Jie Tai Yu,*Face polynomials and inversion formula*, J. Pure Appl. Algebra**78**(1992), no. 2, 213–219. MR**1161345**, https://doi.org/10.1016/0022-4049(92)90099-2**[12]**Jie Tai Yu,*Computing minimal polynomials and the inverse via GCP*, Comm. Algebra**21**(1993), no. 7, 2279–2294. MR**1218498**, https://doi.org/10.1080/00927879308824677

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Additional Information

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

Keywords:
Minimal polynomials,
Galois theory,
product formula,
polynomial automorphisms

Article copyright:
© Copyright 1995
American Mathematical Society