TITLE: SAT and Arithmetic Geometry
QUESTION [16 upvotes]: This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note though that the system is represented in non-trivial manner). The reason it is OK to consider more than one equation is because the conjunction of the conditions $f_i(x_1 ... x_n) = 0$ is equivalent to the single condition $\prod_i (f_i(x_1 ... x_n) + 1) + 1 = 0$.

This reminds of the solution of Hilbert's 10th problem, namely that it is undecidable whether a system of polynomial equations over $\mathbb{Z}$ has roots. Is there a formal relation? Can we use the undecidability over $\mathbb{Z}$ to provide clues why the problem is hard over $\mathbb{F}_2$ (that is, $P \ne NP$)? What is known about decidability and complexity for other rings? In particular, what is known about complexity over $\mathbb{F}_p$ for p prime > 2?
The system of polynomial equations defines an algebraic scheme. Is it possible to find algebro-geometric conditions on this scheme, s.t. something can be told about the complexity of SAT restricted to such schemes?
The solutions of our system of polynomial equations are the fixed points of the Frobenius endomorphism on the corresponding variety over $\bar{\mathbb{F}}_2$. There is a variant of Lefschetz's fixed-point theorem which relates the existence of such points to $l$-adic cohomology. Can this be used to provide some insight on P vs. NP?

REPLY [6 votes]: I'm going to address just one aspect of your question, which has been restated in some of the answers and comments, about whether there is any formal connection between undecidability and computational complexity.
This is certainly a very tempting idea, and you could argue that one of the main reasons to conjecture that the polynomial hierarchy does not collapse (this is a standard generalization of the conjecture that P≠NP) is that the infinitary analogue of the polynomial hierarchy, namely the arithmetical hierarchy, is known not to collapse.  Of course we know, for example from the Baker–Gill–Solovay theorem, that we can't naively carry over all intuitions from computability theory over to complexity theory, but the analogy remains in the back of the minds of many people working in this subject.
Perhaps closer to your question are two papers by Michael Freedman, Limit, logic, and computation and K-SAT on groups and undecidability.  The latter paper in particular notes that 2-SAT is polytime solvable and 3-SAT is NP-complete, and constructs infinitary analogues of these two problems that are respectively decidable and undecidable.  Your observation about the analogy between the MRDP theorem and the hardness of solving systems of equations over $\mathbb{F}_2$ could be thought of as being in the same vein.  Unfortunately, while intriguing, so far these analogies don't seem to have yielded any significant insights into the P≠NP problem.