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.

Thanks in advance.