2
votes
0answers
90 views

Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber.(1996) An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield. ...
4
votes
0answers
153 views

Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...
11
votes
0answers
359 views

Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories.(In 80s) Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: a) Trivial (No ...
17
votes
1answer
689 views

Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any ...
3
votes
0answers
82 views

Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups? The ...
2
votes
0answers
95 views

topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$, call a subset $Z\subseteq U$ $\Gamma$-closed iff it is a closed analytic subset and each of its irreducible components is an ...
1
vote
1answer
327 views

Riemann hypothesis for zeta function of definable sets over finite fields

Hi, Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by ...
3
votes
0answers
140 views

axiomatizing the abelian part of the topological fundamental groupoid functor on algebraic varieties

let Vv be the category of complex algebraic varieties defined over $\bar Q\subset C$, and let $\pi_1^{top}:Vv\longrightarrow Groupoids$ sending a variety V into its (strict) fundamental groupoid ...
2
votes
1answer
150 views

Terminology for system of equations and…

I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...
9
votes
6answers
1k views

Proofs in the same vein as Ax-Grothendieck

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
4
votes
4answers
906 views

Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on. So, I have become interested in ...
11
votes
3answers
994 views

Intuition for Model Theoretic Proof of the Nullstellensatz

I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
9
votes
2answers
678 views

Cohomology Theories on The Stone Space of Complete n-types

Just a random thought here: Can cohomology theories (e.g. sheaf cohomology) on the Stone space $S_n(T)$ (the space of complete n-types) of a first-order theory $T$ tell us anything interesting (e.g. ...
4
votes
2answers
269 views

Determining the field from the topology of the affine space over it.

The inspiration is from another question in which it is remarked that the passage to Zariski topology is a very interesting functor from Commutative Rings to Topological spaces. I am trying to ...
4
votes
1answer
294 views

Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).

Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$. If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...