The Toric Ring and the Toric Ideal Arising from a Nested Configuration

Toric rings and toric ideals are established research objects in combinatorial and computational aspects of commutative algebra. In [1], from a viewpoint of algebraic statistics, the concept of nested configurations is introduced. In the present paper, the toric ring together with the toric ideal arising from a nested configuration will be studied in detail. Let K[t] = K[t1, . . . , td] denote the polynomial ring in d variables over a field K. Recall that a configuration of K[t] is a finite set A of monomials belonging to K[t] such that there exists a nonnegative vector (w1, . . . , wd) ∈ R d ≥0 with ∑d i=1 wiai = 1 for all t1 1 · · · t ad d ∈ A. We will associate each configuration A of K[t] with the homogeneous semigroup ringK[A], called the toric ring of A, which is the subalgebra of K[t] generated by the monomials belonging to A. Let K[X] = K[{xM |M ∈ A}] denote the polynomial ring over K in the variables xM with M ∈ A with each deg(xM) = 1. The toric ideal IA of A is the kernel of the surjective homomorphism π : K[X] → K[A] defined by setting π(xM) = M for all M ∈ A. It is known (e.g., [7, Section 4]) that the toric ideal IA is generated by those homogeneous binomials u− v, where u and v are monomials of K[X], with π(u) = π(v). Now, let A = {t1, . . . , tn} be a configuration of K[t] with the properties that deg tj = r for each 1 ≤ j ≤ n and that, for each 1 ≤ i ≤ d, there is 1 ≤ j ≤ n such that tj is divided by ti. Assume that, for each 1 ≤ i ≤ d, a configuration Bi = {m (i) 1 , . . . , m (i) λi } of a polynomial ring K[u] = K[u (i) 1 , . . . , u (i) μi ] in μi variables over K is given. Then the nested configuration [1] arising from A and B1, ..., Bd is the configuration