Resultants and contour integrals

Abstract

Resultants are important special functions used to describe nonlinear phenomena. The resultant R r1...rn determines a consistency condition for a system of n homogeneous polynomials of degrees r 1 , . . . , r n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.

Complanart of a system of polynomial equations

“…For s = 1, i.e., if A is a linear map, the resultant of A is the determinant of A. Resultants play an ever increasing role in modern mathematics and physics (see, e.g., [1], [2], [9]- [11] for an overview, [12] for applications in physics and engineering, [13], [14] for applications in string theory, and [15], [16] for computational methods). By a nondegenerate map, we mean a map with a nonzero resultant.…”

“…Resultants play an increasing role in modern mathematics and physics. See, for example [2,1,9,10,11] for overview, [12,13] for applications in physics and engineering, [14] for application in string theory and [15,16] for computational methods. By non-degenerate map we mean map with non-vanishing resultant.…”

“…In some particular cases the formula for V j 1 •••j N may be obtained from simple considerations (see for detailed discussion and examples [2]), but we consider now a general method of calculating it. This method uses a Poisson product formula for resultant, see e. g. [1,16]. Add one more homogeneous polynomial g(x) of degree r of the same variables x 1 , • • • , x n .…”

Colored HOMFLY and generalized Mandelbrot set

“…If known, they make the theory of non-linear equations as transparent as it is for linear ones -and the study of resultants and discriminants is the main topic in non-linear algebra [18,[36][37][38][39][40][41][42]. The present knowledge, however, remains restricted -and the problem attracts much less attention and effort than it deserves.…”