Two Way Subtable Sum Problems and Quadratic Gröbner Bases

Hara, Takemura and Yoshida [2] discussed toric ideals arising from two way subtable sum problems and showed that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Gröbner bases. Fix positive integers m and n and T = {(i, j) | 1 ≤ i ≤ m, 1 ≤ j ≤ n}. Let K be a field and K[{ui}1≤i≤m ∪ {vj}1≤j≤n ∪ {w, t}] be the polynomial ring in m + n + 2 variables over K. Given a subset S of T , let RS denote the semigroup ring generated by those monomials uivjw with (i, j) ∈ S and those monomials uivjt with (i, j) / ∈ S. Let K[X] = K[{xi,j}(i,j)∈T ] denote the polynomial ring in mn variables over K. Define the surjective map π : K[X] → RS with π(xi,j) = uivjw if (i, j) ∈ S and π(xi,j) = uivjt if (i, j) / ∈ S. We call the kernel of π the toric ideal of two way subtable sum problems associated with S and denote it by IS. We refer the reader to [1] and [5] for fundamental facts on Gröbner bases and toric ideals. A subset S ⊂ T is called 2× 2 block diagonal if there exist integers r, c such that S = {(i, j) | 1 ≤ i ≤ r, 1 ≤ j ≤ c} ∪ {(i, j) | r < i ≤ m, c < j ≤ n} after an appropriate interchange of rows and columns. A subset S ⊂ T is called triangular if S satisfies the condition (∗) (i, j) ∈ S ⇒ (i, j) ∈ S for all 1 ≤ i ≤ i and 1 ≤ j ≤ j after an appropriate interchange of rows and columns. Hara, Takemura and Yoshida [2] showed that IS is generated by quadratic binomials if and only if S is either 2×2 block diagonal or triangular. Our main theorem is as follows: Theorem. Work with the same notation as above. Then the following conditions