Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective resultant, in the literature, it is called Macaulay's or Dixon's resultant. But a problem arises: these $f_i$ may have a common root in the infinity (in the projective space), but no common root in the affine space.
My question: is there an analogue resultant for the case of affine polynomials?
The example of three linear equations in two variables is typical. The equations $ax+by+c=0$, $dx+ey+f=0$ and $gx+hy+i=0$ have a common root if
Note that we cannot give a single polynomial $R(a,b,c,d,e,f,g,h,i)$ which vanishes if and only if the equations have a common root, because the property of these equations having a common root is not closed. For example, for all $\epsilon \neq 0$, the equations $y=0$, $y+\epsilon x=1$ and $y+2\epsilon x = 2$ have a common root at $(0, 1/\epsilon)$, but the equations $y=0$, $y=1$, $y=2$, obtained in the limit as $\epsilon \to 0$, have no common root. Note that this also shows that no list of equations could do the job.
In general, a constructible set is a subset of an algebraically closed field formed from finitely many polynomial equalities and the boolean connectors AND, OR and NOT. For example, the set $$\{ (a,b,c,d,e,f,g,h,i,x,y) : ax+by+c=dx+ey+f=gx+hy+i=0 \}.$$ A theorem of Chevalley says that the projection of a constructible set onto a subset of its coordinates is again constructible. So the projection of the above set onto the $(a,b,c,d,e,f,g,h,i)$ coordinates must be an constructible set, and I've given a boolean description above.
