# Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.

Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full row rank and $m < n$. Let $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0\}$. For any $u \in \ker A$ vectors $u_+$ and $u_-$ can be defined as follows: $$u_+ = \sum_{u_i > 0} u_i e_i,$$ $$u_- = - \sum_{u_i < 0} u_i e_i.$$ The ideal $I_A = \langle \{ x^{u_+} - x^{u_-} \; | \; u \in \ker A \} \rangle$ is no doubt toric. The well-known way to construct $I_A$ is the following one. Let $L = \{ l^1,...,l^r \}$ $(r = n - m)$ be a basis for $\ker A$ and $$I_L = \langle \{ x^{l^i_+} - x^{l^i_-} \; | \; i = \overline{1,r}\; \} \rangle$$ then $$I_A = I_L : (x_1 \cdots x_n)^\infty.$$ My questions are the following:

1) Is it right that $I_L$ is always $I_A$-primary (i.e. $I_A = \sqrt{I_L}$)? If so is there any relatively easy proof of it? If not what is the case when this statement may fail?

2) Are there any reasonable bounds on the size of a reduced Groebner basis of $I_A$ (for example for grevlex ordering)? Possible bounds in terms of matrix $A$ (or equivalently in terms of its basis $L$) would be much appreciated.

Any hint or reference would help a lot.

The answer to your first question is "No". I think it's almost always No. You can already see this in the well-known twisted cubic example. Here $A = \begin{pmatrix}1& 1& 1& 1\\ 0& 1& 2& 3\\ \end{pmatrix}$ and take, for example, the lattice basis {(1,-2,1,0),(0,1,-2,1)}. The corresponding binomial ideal $(-b^{2}+a c,-c^{2}+b d)$ is not primary. It has two components: $I_L$ and the ideal $(b,c)$
For example consider the famous $n_1\times n_1$ transportation problem, where the matrix $A\in \mathbb Z^{m\times n_1n_2}$ is the linear map that computes the row and column sums of a given $n_1\times n_2$-matrix, then it is well known that the toric Ideal $I_A$ is minimally generated by all $2\times 2$ minors of a generic $n_1\times n_2$ matrix. See "Groebner bases and convex polytopes" by Bernd Sturmfels Example 5.1 and continuation for a more detailed study.
For fixed dimensions $n_1$ and $n_2$ also upper bounds of the total degree of any primitive in the homogeneous toric ideal $I_A$ are know. see "Groebner bases and convex polytopes" corollary 4.15
Further Jesus A. De Loera and Shmuel Onn show in their so called "no hope"-theorem that there is no hope for a general upper bound on the total degree of an element in a minimal Markov basis, even for $3\times n_1n_2$ transportation problems with both dimensions $n_1$ and $n_1$ varying.