TITLE: Can it be decidable for any polynomials to have the intersecting point?
QUESTION [5 upvotes]: Give system of polynomials$$P_1(x_1,x_2,\dots,x_n)=0,$$$$\vdots,$$$$P_k(x_1,x_2,\dots,x_n)=0$$
Can it be decidable for thoses polynomials to have the intersecting point ?

REPLY [3 votes]: Given an ideal $I$ of $k[x_1,\ldots,x_n]$ with $k$ an algebraically closed field of characteristic zero, you can effectively compute a $Gr\ddot{o}bner$ basis for it. Checking whether $Z(I)$ is empty is equivalent to checking whether $1$ is in the  $Gr\ddot{o}bner$ basis (by Hilbert's Nullstellensatz). Checking whether $1$ is in the  $Gr\ddot{o}bner$ basis is a trivial check.
Maybe the same works in characteristic $>0$ as well.
As André Nicolas says the problem is much more difficult over non-algebraically closed fields, even if you restrict to just smooth projective curves.