most recent 30 from http://mathoverflow.net 2013-05-25T08:47:03Z http://mathoverflow.net/feeds/question/81327 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81327/intersecting-the-algebraic-closure-of-independent-elements Drike 2011-11-19T09:08:06Z 2011-11-20T10:38:57Z

<p>$G$ is a group with a simple first order theory $T$ as defined by Shelah, hence equiped with a "nice" notion of independence. $G$ also has generic elements. I write $acl^n(A)$ for the set of elements in $M$ lying in a finite $A$-definable set of size at most $n$, and $acl(A)$ for the union of $acl^n(A)$ when $n$ ranges over all natural numbers. Let $a$ and $b$ be independent elements over $0$. One has </p> <p>$$acl(a)\cap acl(b)=acl(\emptyset)$$</p> <p>In particular, this holds if $a$ and $b$ are independent generic elements of $G$.</p> <p><strong>Question 1 :</strong> If $a$ and $b$ are generic, does the following equality hold? $$acl^n(a)\cap acl^n(b)=acl^n(\emptyset)$$</p> <p>As I expect the answer to question 1 to be no, let me ask :</p> <p><strong>Question 2 :</strong> Is there a constant $k$ (depending only on $T$ and $n$) such that for all independent generics $a$ and $b$ over $0$, one would have $$acl^n(a)\cap acl^n(b)\subset acl^{k.n}(\emptyset)$$</p>