Polarizations of powers of graded maximal ideals

2019/12/09 by Ayah Almousa, Almousa, Ayah, Gunnar Fløystad +3 · 2 citations
Mathematics · #05E40 #13F55 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13F20 #Secondary: 55U10 #math.AC #msc:05E40 #msc:13F20 #msc:13F55 #msc:55U10

paper · pdf · doi:10.48550/arxiv.1912.03898

37 pages, Section 3 contains several conjectures and problems

arxiv created 2020/11/09 · arxiv updated 2020/11/10

Abstract

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x₁,x₂,⋯,x_m)ⁿ of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case m=3 and also in the power two case n=2 the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal defines a simplicial ball.

Cited by

• Monomial polarization and depolarization of abstract simplicial complexes
• Ordinary and symbolic powers of matroids via polarization

Related

• The many polarizations of powers of maximal ideals
• Deformations of polarizations of powers of the maximal ideal
• Polarizations and Hook Partitions
• Polarizations of Artin monomial ideals
• Alternative polarizations of Borel fixed ideals