Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions

Abstract

In this paper we study some problems concerning bigraded ideals. By introducing the concept of bigeneric initial ideal, we answer an open question about diagonal subalgebras and we give a necessary condition for a function to be the bigraded Hilbert function of a bigraded algebra. Moreover, we give an upper bound for the regularity of a bistable ideal in terms of the degrees of its generators

Multigraded regularity: Coarsenings and resolutions

“…In particular, we show how to use reg-num v (M) to determine bounds on the multigraded regularity of M as defined in [11]. As well, we relate reg-num v (M) to the resolution regularity vector of M as defined by [1,14].…”

Linear spaces, transversal polymatroids and ASL domains

“…As L is generic, the initial ideal in(I) is the multigraded generic initial ideal of I with respect to >. Hence in(I) is Borel fixed is the multigraded sense, see [1]. In characteristic 0 this means that if a monomial M is in in(I) and t ij |M then t ik M/t ij is in in(I) as well for all the k > j.…”

Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals

“…See e.g., [13]. Also, for r = 2, G-generic initial ideals and strongly color-stable ideals are considered in [2]. Now we define colored algebraic shifting.…”

“…Let be a set of indices, X = {x τ : τ ∈ } a set of variables and K[X] the polynomial ring over a field K in the set of variables X. Consider the set of variablesX = {x τ, [k] : [1] x τ j , [2] · · · x τ j , There is a nice relationship between polarization and graded Betti numbers. For a finitely generated graded ideal I ⊂ K[X], we define the graded Betti numbers of I by The following nice fact is known: Let I be a monomial ideal of K[x 1,1 , .…”

“…However, pol(I ) and (I ) do not have such a nice relationship when I is not strongly stable. Indeed, if I = x 2 1 , x 2 2 then pol(I ) and (I ) = x 1 x 2 , x 2 x 3 do not have the same Hilbert function.…”