2008/04/29 by Morozov, A., Shakirov, Sh. · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.0804.4632
Resultant Rr1, ..., rn defines a condition of solvability for a system of n homogeneous polynomials of degrees r1, ..., rn 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.