Resultant as Determinant of Koszul Complex

TL;DR

This paper explores the relationship between determinants of linear maps, complexes, and resultants, emphasizing that the resultant of a non-linear map is the determinant of its associated Koszul complex, with implications for theoretical physics.

Contribution

It provides an elementary introduction to determinants of complexes and resultants, highlighting their interrelations and potential significance in future theoretical physics research.

Findings

Resultant of a non-linear map equals the determinant of its Koszul complex

Clarifies the connection between determinants of complexes and resultants

Lays groundwork for applications in theoretical physics

Abstract

A linear map between two vector spaces has a very important characteristic: a determinant. In modern theory two generalizations of linear maps are intensively used: to linear complexes (the nilpotent chains of linear maps) and to non-linear mappings. Accordingly, determinant of a linear map has two generalizations: to determinants of complexes and to resultants. These quantities are in fact related: resultant of a non-linear map is determinant of the corresponding Koszul complex. We give an elementary introduction into these notions and interrelations, which will definitely play a role in the future development of theoretical physics.