Infinitesimal Center Problem on zero cycles and the composition conjecture

TL;DR

This paper proves the composition conjecture for the infinitesimal center problem on zero cycles, establishing a clear criterion for when the displacement function vanishes, with applications demonstrated.

Contribution

It establishes the validity of the composition conjecture for zero cycles, contrasting with the tangential center problem, and introduces the displacement function in this context.

Findings

Displacement function is zero iff the deformation has a composition factor.

The composition conjecture holds for zero cycles.

Applications of the results are provided.

Abstract

We study the analogue of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give examples of applications of our results.