Related papers: Structural decomposition of monomial resolutions
Given any monomial ideal $ I $ minimally generated by $ q $ monomials, we define a simplicial complex $\mathbb{M}_q^2$ that supports a resolution of $ I^2 $. We also define a subcomplex $\mathbb{M}^2(I)$, which depends on the monomial…
Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine…
Answering complex questions often requires multi-step reasoning in order to obtain the final answer. Most research into decompositions of complex questions involves open-domain systems, which have shown success in using these decompositions…
We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective…
We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…
Using the Newton polytope and polyhedron, we study analytic spread and ideal reductions of monomial ideals. We determine a bound for analytic spread based on halfspaces and hyperplanes of the Newton polytope, and we classify basic monomial…
Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…
The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields…
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
Given an ideal $I$ we investigate the decompositions of Betti diagrams of the graded family of ideals $\{I^k \}_k$ formed by taking powers of $I$. We prove conjectures of Engstr\"om and show that there is a stabilization in the…
An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
In this note, we study Betti splittings of cover ideals of bipartite graphs. We prove that if $J \subset \Bbbk [x_1,\dots,x_n]$ is the cover ideal of a bipartite graph then the $x_i$-partition of $J$ is a Betti splitting for any $i$. We…
We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and…
We present an algorithm which computes a cylindrical algebraic decomposition of a semialgebraic set using projection sets computed for each cell separately. Such local projection sets can be significantly smaller than the global projection…
Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its…
A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
We determine a new technique which allows the computation of the arithmetical rank of certain monomial ideals.
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli…