Computing real powers of monomial ideals

2021

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real powers of a monomial ideal generalize the integral closure operation and highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the real powers of a monomial ideal. An important result is that given any monomial ideal I, the function taking real numbers to the corresponding real power of I is a step function whose jumping points are rational. This reduces the problem of determining real powers to rational exponents.

Journal of Pure and Applied Algebra, 2022

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. We introduce a novel family of irreducible powers. Irreducible powers and symbolic powers of monomial ideals are studied by means of the corresponding irreducible polyhedron and symbolic polyhedron respectively. 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 monomial ideal, an asymptotic invariant which can be defined using the symbolic polyhedron, by means of an analogous invariant stemming from the irreducible polyhedron, which we introduce under the name of naive Waldschmidt constant.

Proceedings of the Edinburgh Mathematical Society, 2016

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Mathematische Zeitschrift, 2021

Let I be a homogeneous ideal in a polynomial ring over a field. Let I (n) be the n-th symbolic power of I. Motivated by results about ordinary powers of I, we study the asymptotic behavior of the regularity function reg(I (n)) and the maximal generating degree function ω(I (n)), when I is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences {reg I (n) /n}n and {ω(I (n))/n}n converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal I for which ω(I (n)) and reg(I (n)) are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

We prove that the integral closures of the powers of a squarefree monomial ideal I equal the symbolic powers if and only if I is the edge ideal of a Fulkersonian hypergraph.

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt cons...

Communications in Algebra, 2018

Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.

Proceedings of the American Mathematical Society, 2011

We generalize an example, due to Sylvester, and prove that any monomial of degree d in R[x 0 , x 1 ], which is not a power of a variable, cannot be written as a linear combination of fewer than d powers of linear forms.