Translation of Tschebotarow's "The Problem of Resolvents and Critical Manifolds"

TL;DR

This paper provides an English translation of Chebotarev's work on resolvent problems, linking them to critical manifolds and foundational concepts like resolvent degree and essential dimension in algebraic equations.

Contribution

It introduces Chebotarev's approach to resolvent problems using critical manifolds, clarifying their connection to resolvent degree and essential dimension.

Findings

Resolved the problem of finding minimal-parameter resolvents.

Connected resolvent problems to the geometry of critical manifolds.

Contributed to the theoretical foundation of algebraic resolvent analysis.

Abstract

This is an English translation of "The Problem of Resolvents and Critical Manifolds" by Tschebotarow/Chebotarev. In this article, Chebotarev explains his work on resolvent problems using critical manifolds. The current ideas of resolvent degree and essential dimension arose out of the resolvent problems Chebotarev addresses here. Original abstract by Chebotarev: "This paper is devoted to the study of the problem of resolvents, i.e., the problem of finding the resolvent by a given equation, whose coefficients depend on several independent parameters, the number of parameters in the coefficients being as small as possible. The author connects the problem to the study of critical manifolds in the space of equation parameters."