New and Old Results in Resultant Theory

TL;DR

This paper reviews the historical and recent developments in the theory of resultants, emphasizing explicit formulas that are useful for modern physics applications involving non-linear equations.

Contribution

It provides a comprehensive overview of classical and recent results in resultant theory, highlighting explicit formulas relevant to physics.

Findings

Compilation of explicit formulas for resultants

Discussion of resultants in non-linear and non-Gaussian contexts

Identification of practical formulas for physics research

Abstract

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.