Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus (Q404013): Difference between revisions

For a commutative \(\mathbb{Q}\)-algebra \(R\) let \(R^N\) denote the affine \(N\)-space \(\mathrm{Spec}\;R[X_1 , X_2 , \dots, X_N ]\). Let \(f : R^2 \to R^2\) be a polynomial mapping given by a pair of polynomials \(p, q\). Define \(\mathrm{deg}\;f = \max {\mathrm{deg} p, \mathrm{deg} q}\). In this paper the author considers triangular automorphisms \(f\) of \(R^2 \), i.e. those for which \(p\) does not involve \(X_2\). Let \(f^n\) denote the \(n\)th iterate of \(f\) for any integer \(n\). If further the Jacobian determinant of \(f\) is \(1\), then the main result of the paper says that deg \(f^n \leq d^2 - d + 1\). In particular, \(\mathrm{deg}\;f^n\) is bounded above as \(n\) varies over all integers. The author introduces a weighted nilpotency index, \(\nu(f )\), for a polynomial automorphism with determinant one, \(f : R^N \to R^N \). He gives a bound for \(\mathrm{deg} f^n\) in terms of \(N, d, \nu(f )\) for all integers n. For \(n = -1\) a formula found by S. S. Abhyankar and the reviewer in 1974 for the formal inverse of \(f\) (based on an old result of Lagrange) is used. The estimate for deg \(f^{-1}\) was found by \textit{P. van Rossum} around 2001 [Tackling problems on affine space with locally nilpotent derivations on polynomial rings, Ph.D. thesis, University of Nijmegen (2001)].