Weighting representation of Simis monomial ideals (Conjecture 5.7)

Establish that for any Simis monomial ideal I in S = K[t1, ..., ts] with no embedded primes and having a minimal irreducible decomposition, there exist a Simis squarefree monomial ideal J ⊂ S and a standard linear weighting w such that I equals the weighted monomial ideal of J under w, i.e., I = J_w.

Background

The paper introduces weighted monomial ideals via linear weightings and studies how algebraic properties behave under standard weightings. Known results show that many properties, including normality and the Simis property in certain contexts, are preserved when passing to weighted ideals.

Motivated by these connections, the authors conjecture that Simis monomial ideals with minimal irreducible decomposition and no embedded primes arise as weighted versions of Simis squarefree monomial ideals, providing a structural reduction from general Simis ideals to the squarefree case. They present supporting evidence and examples, while noting conditions essential for the conjecture’s validity.