MAPLE---procedures for finding the resultants of systems of nonlinear algebraic equations (Q2716176)

Let NEWLINE\[NEWLINEf_{j}(z)\equiv z^{k_{j}}_{j}+Q_{j}(z) = 0 j=1,\dots ,n,\tag{1}NEWLINE\]NEWLINE where NEWLINE\[NEWLINEz=(z_{1},\dots ,z_{n})\in \mathbb{C}^{n};\;Q_{j}(z)\in \mathbb{C}[z]\text{ and } \deg Q_{j}(z) < k_{j},NEWLINE\]NEWLINE be a system of nonlinear algebraic equations. NEWLINENEWLINENEWLINEIn the paper the author describes a structure of the program on MAPLE (V 5.2) implementing a modified method of elimination of unknowns for the system (1) and slightly more general systems. This method was developed [see \textit{V. I. Bykov, A. M. Kytmanov} and \textit{M. Z. Lazman}, Elimination methods in a computer algebra of polynomials. Novosibirsk, Nauka (1991; Zbl 0743.68048)] on the basis of the multidimensional theory of residues (multidimensional logarithmic residue, residue of Grothendieck etc.). It allows to represent various symmetric functions from the solutions of such systems as (1) through regular functions from coefficients of this system. For example, when \(n = 2\) system (1) looks like NEWLINE\[NEWLINEf_{1} \equiv z_{1}^{3} + A_{1}z_{1}^{2} + B_{1}z_{1}z_{2} + C_{1}z_{2} ^{2} + D_{1}z_{1} + E_{1}z_{2} + F_{1} = 0 NEWLINE\]NEWLINE NEWLINE\[NEWLINE f_{2} \equiv z_{2}^{3} + A_{2}z_{1}^{2} + B_{2}z_{1}z_{2} + C_{2}z_{2} ^{2} + D_{2}z_{1} + E_{2}z_{2} + F_{2} = 0.NEWLINE\]NEWLINE The resultant of this system on a variable \(z_{1}\) looks like NEWLINE\[NEWLINE RES = z_{1}^{9} + G_{1}z_{1}^{8} + \dots + G_{8}z_{1} + G_{9}NEWLINE\]NEWLINE The author has obtained (with the help of above program) the values of coefficients \(G\) in a symbolic form NEWLINE\[NEWLINEG_{1}=3A_{1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{aligned} &G_{9} = C_{1}C_{2}^{2}F_{1}^{2}-2C_{1}^{2}C_{2}F_{1}F_{2} - C_{1}C_{2}E_{1}E_{2}F_{1} + C_{1}C_{2}E_{1}^{2}F_{2} - C_{2}E_{1}F_{1}^{2}\\ &+ C_{1}^{3}F_{2}^{2} + C_{1} ^{2}E_{2}^{2}F_{1} - 2C_{1}E_{2}F_{1}^{2}+ 3C_{1}E_{1}F_{1}F_{2} + E_{1}^{2}E_{2}F_{1} - E_{1} ^{3}F_{2} + F_{1}^{3}.\end{aligned}NEWLINE\]NEWLINE{}.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].