On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = − y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.

Key Words:

• Invariant field
• Invariant algebra
• Unfaithfull degree
• Algebraically independent polynomials
• Intersection multiplicity
• Jacobians

AMS Subject Classification:

• Primary 12F10
• 13B25
• Secondary 13A50

Acknowledgments