I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand.
I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ has any solutions is undecidable (Hilbert's 10th problem). Since the integers form a ring, $\mathbb{Z}[\vec{x}]$ (the polynomials over $\mathbb{Z}$ with coefficients $\vec{x}$) is also a ring. So the question can be restated as asking whether the variety of $\langle P \rangle$ is empty or not. But $\mathbf{V}(\langle P \rangle) = \mathbf{V}(G(\langle P \rangle))$
where $\mathbf{V}(I)$ is the variety of the ideal $I$, and $G(I)$ is a Groebner basis for $I$. So we can check whether $\mathbf{V}(\langle P \rangle)$ is empty by checking whether $G(\langle P \rangle) = G(\langle 1 \rangle)$. But that would give us a decision procedure for solving Diophantine equations.
I'm assuming that I've gone wrong either:
- Assuming that I can compute a Groebner basis for $\langle P \rangle$ (since the integers don't form a field, I don't know whether this is legit)
- Assuming that Groebner bases have the same properties (like being canonical) if they're done over an arbitrary, non-field base ring
- Something else?
As I said, I'm just getting started looking at the whole field, so any pointers as to huge conceptual mistakes I'm making would be hugely appreciated.
Thanks.