An Approach to Determine the Resultant of Two Entire Functions

Abstract

Classical recurrent Newton’s identities give relations between sums of powers of the roots of a polynomial and the coefficients of this polynomial (see, e.g., [1–3]). These formulas can be obtained with the use of the Cauchy integral formula [4, Ch.1]. This fact allows us to expand the class of functions for which these recurrent formulas are valid. Namely, for the class of entire functions of finite order of growth one can obtain relations between the coefficients of a Taylor expansion of a given function and sums of negative powers of zeros of the function [4, Ch.1]. Using the methods of complex analysis, we introduce the concept of the resultant for an entire function and an entire function with finite number of zeros and establish its properties. The proposed approach can be useful, for example, in studies of equations of chemical kinetics where exponential polynomials arise [5, 6]. It also allows us to apply this approach to the elimination of unknowns from systems of non-algebraic equations on the basis of the Zel’dovich-Semenov scheme [7]. Let us consider two polynomials f and g. The classical resultant R(f, g) can be defined in several ways: a) by using the Sylvester determinant (see, e.g., [1–3]); b) by virtue of the product formula R(f, g) = ∏