Given any admissible finite O-sequence h: 1,n,...,h(s-1), we introduce on the set of zero-dimensional Borel ideals in the polynomial ring in n variables (over a field of characteristic 0), having h as Hilbert function, an equivalence relation and a poset structure on the quotient. For every n, the Lex-segment ideal with Hilbert function h gives the unique maximal element of this poset which, for n grater or equal to 4, has several different minimal elements. For n=3 the poset structure is actually a lattice structure and we construct the generalized rev-lex-segment ideal which gives the unique minimal element of this lattice.
Borel Ideals in Three Variables
MARINARI, MARIA GRAZIA;RAMELLA, LUCIANA
2006-01-01
Abstract
Given any admissible finite O-sequence h: 1,n,...,h(s-1), we introduce on the set of zero-dimensional Borel ideals in the polynomial ring in n variables (over a field of characteristic 0), having h as Hilbert function, an equivalence relation and a poset structure on the quotient. For every n, the Lex-segment ideal with Hilbert function h gives the unique maximal element of this poset which, for n grater or equal to 4, has several different minimal elements. For n=3 the poset structure is actually a lattice structure and we construct the generalized rev-lex-segment ideal which gives the unique minimal element of this lattice.
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