Inverse degree of an affine space triangular automorphism

Abstract:

For any $\mathbb {Q}$-algebra $R$ and any triangular automorphism $f: R^N\to R^N$ with Jacobian one on the affine space, we show that $\deg (f^{-1})$ is bounded from above by a constant depending only on $N$ and $\deg (f)$. This is seen as a generalization of a result by Furter on the affine plane. Our proof uses (a version of) Furter’s estimate on nilpotency indices and Abhyankar–Gurjar’s formal inversion formula. It follows that when the Jacobian of a triangular automorphism $f: R^N\to R^N$ is not necessarily equal to one, $\deg (f^{-1})$ is bounded from above by a constant depending only on $N$, $\deg (f)$ and $\deg (1/\operatorname {Jac}(f))$.