MONOMIAL IDEALS WITH TINY SQUARES AND FREIMAN IDEALS

We provide a construction of monomial ideals in R = K[x, y] such that
µ(I
2
) < µ(I), where µ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in
the ring R, we generalize the definition of a Freiman ideal which was introduced in J. Herzog,
G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case
of this characterization leads to some further investigations on µ(I
k
) that generalize some
results of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi
Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019).