Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus

Abstract

Let f be a triangular automorphism of the affine N-space of degree d with Jacobian 1 over a ${\mathbb{Q}}$-algebra R. We introduce a weighted nilpotency index ν(f) for f, and give a bound of deg(f ⁿ) in terms of N, d and ν(f) for all ${n \in \mathbb{Z}}$. When N = 2, our formula, combined with computation of the Hilbert series of certain graded algebras, yields the estimate deg(f ⁿ) ≤ d ² − d + 1 for all ${n \in \mathbb{Z}}$. If n varies through all integers, this estimate turns out to be sharp and is related, somewhat unexpectedly, to the Schubert calculus on the Grassmannian G(d − 1, 2d − 2). Numerical computation for small degrees suggests that this estimate restricted to the inverse degree (i.e. n = −1) is also sharp if d ≥ 3.