Radicals of binomial ideals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:43:08Z http://mathoverflow.net/feeds/question/47855 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47855/radicals-of-binomial-ideals Radicals of binomial ideals Timothy Wagner 2010-12-01T03:31:27Z 2010-12-29T15:22:14Z <p>Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,...,m_t)$, the radical of $M$ itself is monomial and can be written as, $Rad(M)=(\sigma(m_1),\sigma(m_2),...,\sigma(m_t))$ where $\sigma(x_1^{a_1}x_2^{a_2}...x_n^{a_n})$ is the product of indeterminates $x_i$ s.t. $a_i\geq 1$. </p> <p>A binomial ideal in $R$ is generated by binomials. I was wondering if we have similar theorems for the case of binomial ideals where we can write down a generating set for the radical by just knowing a generating set of the ideal. Eisenbud and Sturmfels, in their monumental paper on binomial ideals, showed that the radical itself is binomial. I am especially interested in finding generators for radical of binomial ideals in the case where char$(k)=0$ (or even when $k=\mathbb{C}$) and what kind of binomials generate radical binomial ideals. </p> <p>Becker, Grobe and Niermann discuss the case of zero dimensional binomial ideals. Ojeda and Sanchez prove some results for radicals of lattice (binomial) ideals. I have also seen some results in positive characteristic, but they are not relevant to my research. </p> http://mathoverflow.net/questions/47855/radicals-of-binomial-ideals/47900#47900 Answer by Thomas Kahle for Radicals of binomial ideals Thomas Kahle 2010-12-01T13:38:04Z 2010-12-01T13:38:04Z <p>The minimal primes (and sometimes also their intersection) can be computed relatively quickly (compared to primary decomposition) using Algorithm 4 of <a href="http://arxiv.org/pdf/0906.4873v3" rel="nofollow">http://arxiv.org/pdf/0906.4873v3</a>. I've looked at binomial ideals for some time and I doubt that there is an easy way to see the generators of the radical.</p>