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. $a_i\geq 1$.

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.

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.

• Did you check this out: front.math.ucdavis.edu/1009.2823? – Hailong Dao Dec 1 '10 at 4:22
• @Hailong: Thanks for the link. I hadn't seen this before, though curiously, the word "radical" does not appear even once in the article. I'll see if the primary decomposition methods are of any help. – Timothy Wagner Dec 1 '10 at 6:32
• Have you seen this: arxiv.org/abs/alg-geom/9401001? – J.C. Ottem Dec 1 '10 at 9:24
• @Tymothy: the radical is the intersection of all minimal primes, so in some sense you only need to know the minimal primes. – Hailong Dao Dec 1 '10 at 13:59
• @Hailong: Yes I understand that. But I am looking for a more concrete description of the ideals in terms of generators rather than as intersection of several prime ideals. – Timothy Wagner Dec 1 '10 at 21:34