Conditions for Generic Initial Ideals to be Almost Reverse Lexicographic

2007/07/10 by Young Hyun Cho, Cho, Young Hyun, Jung Pil Park +1 · 1 citation
Mathematics · #13A02 #13C05 #13D40 #13E10 #13P10 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A02 #msc:13C05 #msc:13D40 #msc:13E10 #msc:13P10

paper · pdf · doi:10.48550/arxiv.0707.1365

10 pages

arxiv created 2007/07/16 · arxiv updated 2009/12/01

Abstract

Let I be a homogeneous Artinian ideal in a polynomial ring R=k[x₁,...,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 ≤ 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=(x₁^d₁,...,x_n^d_n), we prove that \gin(I) is almost reverse lexicographic if d_i > ∑_j=1^i-1 d_j - i + 1 for each i ≥ 4. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fröberg conjecture.

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• An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

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