On inverse degrees of quadratic triangular automorphisms over ℚ-algebras

ABSTRACT

Let f be a quadratic triangular automorphism of the affine N-space with Jacobian 1 over a ℚ-algebra R. For any N≥4, we show that $deg\left(f^{-1}\right)\le 3\cdot 2^{N-1}-(N+1)$. We also give an example of f with $deg\left(f^{-1}\right)=(7∕4)\cdot 2^{N-1}$, using easy properties of the ring that is isomorphic to the cohomology ring of a full flag variety.

KEYWORDS:

• Automorphism
• inverse degree
• non-reduced ring
• polynomial ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 08A35
• 13B25
• 14J50
• 14R10

Acknowledgements

The author thanks Professor Toshiaki Maeno for the inputs on Proposition 4.1.