Can it be decidable for any polynomials to have the intersecting point?

Question

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 ?

Answer 1

We answer the question for polynomials with rational coefficients.

If we are interested in whether there is a solution over the reals, or over the complex numbers, the answer is yes.

More strongly, the first-order theory of each is decidable. The decidability of the first-order theories was first proved by Tarski.

If we are interested in integer solutions, the answer is no, by the negative solution to Hilbert's Tenth Problem. The final piece in the proof was provided by Matiyasevich.

If we are interested in rational solutions, whether or not there exists a decision procedure is an open problem.

Answer 2

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.