Bar code for monomial ideals

Abstract

Aim of this paper is to count 0-dimensional stable and strongly stable ideals in 2 and 3 variables, given their (constant) affine Hilbert polynomial.To do so, we define the Bar Code, a bidimensional structure representing any finite set of terms M and allowing to desume many properties of the corresponding monomial ideal I, if M is an order ideal. Then, we use it to give a connection between (strongly) stable monomial ideals and integer partitions, thus allowing to count them via known determinantal formulas. P(n, k) = 0 for k > n P(n, n) = 1 P(n, 0) = 0We define now the notion of plane partition.

Bar Code and Janet-like division

“…In this section, referring to [5,4], we summarize the main definitions and properties about Bar Codes, which will be used in what follows. First of all, we recall the general definition of Bar Code.…”

“…We outline now the construction of the Bar Code associated to a finite set of terms. For more details, see [4], while for an alternative construction, see [5]. First of all, given a term t = x…”

“…It is also possible to associate a finite set of terms M B to a given Bar Code B. In [5] we first give a more general procedure to do so and then we specialize it in order to have a unique set of terms for each Bar Code.…”

“…the e-list of B (1) 3 is e(B (1) 3 ) := (0, 1, 0); the bars involved in its computation as stated in Definition 7 are those highlighted in blue in the above picture. ♦ Proposition 10 (Admissibility criterion, [5]). A Bar Code B is admissible if and only if, for each 1-bar B (1) j , j ∈ {1, ..., µ(1)}, the e-list e(B (1) j ) = (b j,n , ...., b j,1 ) satisfies the following condition: ∀k ∈ {1, ..., n} s.t.…”

“…Bar Codes are combinatorial objects encoding many properties of monomial ideals. In [5], they have been employed to count zerodimensional (strongly) stable monomial ideals in 2 and 3 variables with affine Hilbert polynomial p ∈ N, setting a bijection between such ideals and some particular partition of the integer p and then counting these partitions using determinantal formulas. In [7], instead, they are the main tool to compute the Groebner escalier of zerodimensional radical ideals given their variety, without passing through the (usually inefficient) Groebner bases' computation.…”