Combinatorial characterization of Simis ideals

Characterize combinatorially the class of Simis monomial ideals—namely, monomial ideals I in a polynomial ring S = K[t1, ..., ts] that satisfy I^{(n)} = I^n for all integers n ≥ 1—by identifying necessary and sufficient combinatorial conditions under which symbolic and ordinary powers coincide.

Background

The paper studies symbolic versus ordinary powers of monomial ideals, introducing Simis ideals as those for which equality holds for all powers. While complete characterizations exist for several specific families—such as squarefree monomial ideals (edge ideals of clutters), ordinary edge ideals of graphs, certain generalized edge ideals, ideals of covers of graphs, and edge ideals of weighted oriented graphs—the general problem remains unresolved.

The authors emphasize the difficulty of providing a combinatorial characterization of Simis ideals in full generality, noting that the situation is even more challenging if symbolic powers are defined using all associated primes.