Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphisms between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.
The Newton-Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have generalized this identity by replacing the elementary symmetric polynomials with monomial symmetric polynomials.
We give explicit formulas for the Bhattacharya function of m-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.
We give a combinatorial description of the roots of the Bernstein-Sato polynomial of a monomial ideal using the Newton polyhedron and some semigroups associated to the ideal.
Given the monomial ideal I=(x_1^{α_1},...,x_{n}^{α_{n}})\subset K[x_1,...,x_{n}] where α_{i} are positive integers and K a field and let J be the integral closure of I . It is a challenging problem to translate the question of the normality of J into a question about the exponent set Γ(J) and the Newton polyhedron NP(J). A relaxed version of this problem is to give necessary or sufficient conditions on α_1,...,α_{n} for the normality of J. We show that if α_{i}ε{s,l} with s and l arbitrary positive integers, then J is normal.
Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result on splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes.
This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast simplification and improvement of several statements concerning rational maps and their images. A ring-homomorphism like map is introduced that allows for a close comparison between the respective graphs of a rational map and its Newton dual counterpart.
Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomi…
The notion of quasi $f$-ideals was first presented in $[14]$ which generalize the idea of $f$-ideals. In this paper, we give the complete characterization of quasi $f$-ideals of degree greater or equal to $2$. Additionally, we show that the property of being quasi $f$-ideals remains the same after taking the Newton complementary dual of a squarefree monomial ideal $I$ provided that the minimal generating set of $I$ is perfect.
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.
We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For instance when the ideal is of finite codimension, invariants like integral closure and Hilbert-Samuel multiplicity were already combinatorially determined in the very special cases of monomial or non degenerate ideals, using the Newton polygon of the ideal. With our approach, we can generalize these results to arbitrary ideals. In particular the Rees valuati…
For which monomial supports do most polynomials generate a prime ideal? We give necessary and sufficient conditions for the radical of the ideal to be prime over an algebraically closed field. In characteristic zero, the same conditions give primeness. As an application we show that under the same combinatorial conditions on Newton polytopes, the stable intersection of tropical hypersurfaces is connected through codimension one.
Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is chordal if and only if $G$ admits a perfect elimination order. It is known by Fröberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only if the complementary graph $G^c$ of $G$ is chordal. In this article, we discuss some algebraic consequences of Dirac's theorem in the theory of homological shift ideals of edge ideals. Recall that if $I$ is a monomial ideal, $\mbox{HS}_k(I)$ is the monomial ideal generated by the $k$th multigraded shifts of $I$. We prove that $\mbox{HS}_1(I)$ has linear quotients, …
In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of $G$ has a linear resolution), we show that there exists a reduced Gröbner basis $\mathcal{G}$ of the toric ideal of $G$ such that all the monomials in the support of $\mathcal{G}$ are squarefree. Finally, we show (using work by Herzog and Hibi) that if $I$ is a monomial ideal generated in degree 2, then $I$ has a linear resolution if and only if all powers o…
