Question: Solving for coefficients of polynomial equations

Suppose I have a set of P polynomial equations in terms of N variables which are the coefficients of the equations. The equations are generated by the main program and are not known beforehand.

Example (P = 2):

e1 := c1 + 2*c2 + (c3+3*c4)*x*y + (c5+c6)*y^2 +(c8-2)*y^3*x^2 = 0;
e2 := (c4 + 2*c5 + c7)*x + (c9+ 2*c3+5*c4)*x^2 + (2*c7+5*c5+c6)*y^2*x^3 =0;

Because e1 and e2 must be zero for all x and y this implies that all the coefficients must be zero:

c1 + 2*c2 = 0
c3+3*c4 = 0 etc

This gives M linear equations in terms of N unknowns with N > M.

Given the P equations is there a way to automatically set up the M equations and solve for the N unknowns? In some cases it is possible that there are specific values of some of the c's eg c8 = 2 otherwise some of the c's will be expressed in terms of the other c's eg c1 = - 2*c2.

Thanks.

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