Resultant of system with 3 polynomials and 3 variables Let us say I have a system of 3 polynomials, f1(x,y,z), f2(x,y,z), f3(x,y,z). How to find the resultant of these 3 polynomials? What I mean is: is there any special method to do this? Does the Macaulay resultant apply to this kind of problem?
 A: Are your polynomials homogeneous? If so, then yes, the Macauly resultant is what you want. The Macauly resultant of $f_1,f_2,f_3$ is a single polynomial in the coefficients of $f_1,f_2,f_3$ that will vanish if and only $f_1,f_2,f_3$ have a common zero in $\mathbb{P}^2(K)$ (for an algebraically closed field $K$). This even works if $K$ has characteristic $p$.  
It follows in general from elimination theory that there is some polynomial ideal of the coefficients whose vanishing implies the existence of a common zero, but I don't know an intrinsic reason why the case of $n$ homogeneous polynomials in $n$ variables gives a principal ideal.
A: See  HAL : hal-00912907, version 1 , An Introduction to Trägheitsformen. An explicit method is developed there for particular cases.
A: (Please could you put this answer together with the preceding one. I do not know, myself, how it works.)
     There is An algorithm to compute resultants, by Marc Chardin , published in Effective Methods in Algebraic Geometry, Edit.Teo Mora, Carlo Traverso (Birkhäuser). The case of three polynomials is studied explicitely :
"Abstract: We here give a method to calculate the resultant of three polynomials in terms of a square- free decomposition and resultants of two polynomials. After that, we show how the subresultant algorithm enables us to avoid many calculations.
In the last part, we study the possible extension to the general case of n homogeneous polynomials in n variables."
