Workshop Polynomial Automorphisms

During the two talks I will present a survey on recent developments concerning the following problems coming from work of Belov-Kontsevich, Shestakov-Umirbayev, Adjamagbo-Van den Essen, Dumas-Andruskiewitsch, Launois-Lopes.

• Equivalence of the Jacobian, Dixmier and Poisson conjectures.
• Belov-Kontsevich conjecture and related results on the automorphism group of the Weyl algebra.
• Dumas-Andruskiewitsch conjecture and related results on the automorphism group of the algebra U_q⁺(g), the quantum version of the positive part of the simple Lie algebra g.

• Lecture 1. The Tame Generators Problem.

  In this lecture we introduce the group of polynomial automorphisms of affine n-space and wonder what are the generators of this group. Just as invertible matrices over a field are products of elementary matrices one wonders if invertible polynomial maps are also finite compositions of so-called elementary polynomial maps. This problem, called the Tame Generators Problem, has a long history and was recently solved by Shestakov and Umirbaev. They showed that in dimension three there exist non-tame automorphisms. We discuss both the history of the problem and give some idea of its solution.

• Lecture 2. Stable tameness.

  In this lecture we will discuss some very recent results due to Berson, Wright and the lecturer which assert that all known non-tame automorphisms become tame after adding a finite number of variables.

Mori Theory is a powerful tool to understand the birational geometry of projective varieties of arbitrary dimension. In this framework, Corti (1995) gave a rigorous proof of the so called Sarkisov Program, that gives an algorithm to decompose any birational map between Mori fibrations into elementary links. For instance, we can apply this algorithm to any polynomial automorphism of Cⁿ, viewed as a birational map from Pⁿ to Pⁿ.

To be of real interest, such an algorithm should preserve the affine part, that is, all the blow-ups and blow-downs involved in the decomposition should take place on the divisor at infinity. To obtain such a property, it is natural to work with logarithmic Mori Theory, i.e. with pairs (X,B) where X is projective and B is a boundary divisor. Matsuki and Bruno (1997) worked out a Logarithmic Sarkisov Program, but it seems that this algorithm is still perfectible.

• Lecture 1.

  In the first lecture we consider the case of surfaces. We describe how the algorithm of Corti-Matsuki-Bruno works on some simple examples (triangular automorphisms of C² or of the affine quadric surface), and explain how we can get a more satisfying algorithm.

• Lecture 2.

  In the second lecture we discuss the case of dimension 3, focusing on the very basic example of quadratic birational maps from P³ to P³ with inverse of degree 3 (the polynomial automorphism (x,y,z)→ (x+yz, y+z²,z) is such a map).

(joint work with Adrien Dubouloz)