Does a variety contain a cartesian product of two curves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:48:24Z http://mathoverflow.net/feeds/question/66895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66895/does-a-variety-contain-a-cartesian-product-of-two-curves Does a variety contain a cartesian product of two curves? Boris Bukh 2011-06-04T14:13:06Z 2012-07-05T09:12:40Z <p>We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. First, is there an algorithm to decide this? Second, is it true that if $V$ is of degree $d$ and does contain a product of the form $C_1\times C_2$, then $V$ contains a product of the form $C_1\times C_2$ with $\deg C_1,\deg C_2\leq f(d,n)$, for some function $f$?</p> http://mathoverflow.net/questions/66895/does-a-variety-contain-a-cartesian-product-of-two-curves/101384#101384 Answer by Boris Bukh for Does a variety contain a cartesian product of two curves? Boris Bukh 2012-07-05T09:12:40Z 2012-07-05T09:12:40Z <p>Yes, there is such an algorithm. There is an effectively computable constant $N$ such that if $V$ contains a product $S\times T$ where $S,T$ are $N$-point sets, then $V$ contains product of two curves. It is actually true even in the semialgebraic setting. The result is Theorem 1.9 from <a href="http://arxiv.org/abs/1207.0705" rel="nofollow">http://arxiv.org/abs/1207.0705</a></p> <p>[ Apologies for answering my own question with a reference to my own paper. When I asked the question, I did not know the answer. ]</p>