Analogue of the identity Log Det = Trace Log for resultants

Abstract

Resultant $R_{r_1,...,r_n}$ defines a condition of solvability for a system of $n$ homogeneous polynomials of degrees $r_1,...,r_n$ in $n$ variables, just in the same way as determinant does for a system of linear equations. Because of this, resultants are important special functions of upcoming non-linear physics and begin to play a role in various topics related to string theory. Unfortunately, there is a lack of convenient formulas for resultants when the number of variables is large. To cure this problem, we generalize the well-known identity Log Det = Trace Log from determinants to resultants. The generalized identity allows to obtain explicit polynomial formulas for multidimensional resultants: for any number of variables, resultant is given by a Schur polynomial. We also give several integral representations for resultants, as well as a sum-over-paths representation.

24 pages, 2 figures

References in corpus (6)

• Strings, Black Holes, and Quantum Information
• Hidden symmetries of the Nambu-Goto action
• Algebraic Geometry of Discrete Dynamics. The case of one variable
• Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II. Polygons which admit an inscribed circle
• Higher Nilpotent Analogues of A-infinity Structure
• Higher discriminants of polynomials

Cited by in corpus (2)

• Generation of Matrix Models by W-operators
• Resultants and Contour Integrals