Newton Complementary Duals of -Ideals

Abstract

A square-free monomial ideal $I$ of $k[x_{1},\ldots ,x_{n}]$ is said to be an $f$ -ideal if the facet complex and non-face complex associated with $I$ have the same $f$ -vector. We show that $I$ is an $f$ -ideal if and only if its Newton complementary dual $\widehat{I}$ is also an $f$ -ideal. Because of this duality, previous results about some classes of $f$ -ideals can be extended to a much larger class of $f$ -ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for $f$ -vectors of simplicial complexes.

Characterization and Newton Complementary Dual of Quasi $f$-Ideals

“…Then the primary decomposition of I is (x 1 , x 3 ) (x 1 , x 4 ) (x 1 , x 5 ) (x 2 , x 4 ) (x 2 , x 5 ) (x 4 , x 5 ). The facet and non-face complexes of I are (1,5,8,4) and f (δ N (I)) = (1,5,10,6). Thus I is a quasi f -ideal with type (0, 0, 2, 2).…”

“…Any squarefree monomial ideal I ⊂ R can be associated to two different simplicial complexes over the finite set of vertices, denoted by δ F (I) and δ N (I), called the facet complex of I and the non-face complex (or Stanley-Reisner complex) of I respectively. The f -vectors of these two simplicial complexes δ F (I) and δ N (I) have the accompanying prospects : 1) f (δ F (I)) = f (δ N (I)) or 2) f (δ F (I)) = f (δ N (I)) but dim(δ F (I)) = dim(δ N (I)) or 3) f (δ F (I)) = f (δ N (I)) but dim(δ F (I)) = dim(δ N (I)) A squarefree monomial ideal I ⊂ R with property that mention in (1) is called an f -ideal of the polynomial ring R. This notion has been studied for various properties of f -ideals relevant to combinatorial commutative algebra in the papers [3], [4], [11], [12], and [13]. If we look at (1) and (2) collectively, there is one thing common, i.e, dimensions of both δ F (I) and δ N (I) are same.…”

Quasi f-Ideals

“…Later on, the notion of f -graphs (in [13]) and f -simplicial complex (in [14]) was introduced. These notions have been studied for it various properties in the papers [1], [2], [5], [10], [11], [12], [13], [14] and [15]. In Computational Algebraic Geometry and Commutative Algebra, the notion of Hilbert polynomial and Hilbert series are very useful and important invariants of any finitely generated standard graded algebra over some field.…”

“…For example, consider the ideal I = x 1 x 2 , x 3 x 4 , x 1 x 3 x 5 , x 2 x 4 x 5 in the polynomial ring R = k[x 1 , x 2 , x 3 , x 4 , x 5 ]. This ideal I is f -ideal (see [11]); the common f -vector of the facet complex and the non-face complex of I is (5,8,2). Then, by [16,Theorem 6.7.2], the Hilbert series of R/I is equal to…”

“…However, in the same ring, the ideal J = x 1 x 2 x 4 , x 1 x 2 x 5 , x 1 x 4 x 5 , x 2 x 3 x 5 , x 3 x 4 x 5 is not f -ideal because the f -vector of its facet complex is (5, 9, 10) and the f -vector of its Stanley-Reisner complex is (5,10,10). Although J is not f -ideal, yet we can still express the Hilbert series of R/J in terms of the f -vector of its facet complex in the following manner:…”