New and Old Results in Resultant Theory

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.

50 pages, 15 figures

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• Introduction to Integral Discriminants
• The resultant on compact Riemann surfaces
• Algebraic Geometry of Discrete Dynamics. The case of one variable
• Resultants and Contour Integrals
• Analogue of the identity Log Det = Trace Log for resultants
• Non-Linear Algebra and Bogolubov's Recursion
• Complanart of polynomial equations

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