Changements de variable pour un th`eme.

TL;DR

This paper investigates how the invariants of 'thema' structures, introduced earlier, behave under variable changes, showing stability of some invariants but not the isomorphism class, and establishing holomorphic transformation of families.

Contribution

It demonstrates the stability of fundamental invariants under variable change and proves that holomorphic families of thema transform holomorphically.

Findings

Bernstein polynomial invariants are stable under variable change

Principal parameters are stable under variable change

Isomorphism class of a thema is not generally stable

Abstract

We study the behaviour of the notion of "thema", introduced in our previous article [B.09b], by a change of variable. We show not only that the fundamental invariants of such a thema, corresponding to the Bernstein polynomial, are stable by a change of variable, but also other numerical invariants called principal parameters. \\ We show on a rank 3 example that nevertheless the isomorphism class of a thema is not stable in general by a change of variable. We conclude in proving that a change of variable transforms an holomorphic family of thema in an holomorphic family. This implies that non principal parameters change holomorphically.