2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134109Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\gin(I)$ with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fröberg conjecture. And for the case $\Codim I \le 3$, we show that $R/I$ has the strong Lefschetz property if and only if $\gin(I)$ is almost reverse lexicographic. Finally for a monomial complete intersection Artinian ideal $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that $\gin(I)$ is almost reverse lexicographic if $d_i > \sum_{j=1}^{i-1} d_j - i + 1$ for each $i \ge 4$. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fröberg conjecture.10 pagesCommutative Algebra13A02; 13C05; 13D40; 13E10; 13P10text