Set Of Monomials Research Articles

Divsets, numerical semigroups and Wilf’s conjecture

Let S ⊆ N be a numerical semigroup with multiplicity m = min ( S ∖ { 0 } ) and conductor c = max ( Z ∖ S ) + 1 . Let P be the set of primitive elements, i.e. minimal generators, of S, and let L be the set of elements of S which are smaller than c. Wilf’s conjecture (1978) states that the inequality | P | | L | ≥ c must hold. The conjecture has been shown to hold in case | P | ≥ m / 2 by Sammartano in 2012, and subsequently in case | P | ≥ m / 3 by the author in 2020. The main result in this paper is that Wilf’s conjecture holds in case | P | ≥ m / 4 when m divides c. The proof uses divsets X, i.e. finite divisor-closed sets of monomials, as abstract models of numerical semigroups, and proceeds with estimates of the vertex-maximal matching number of the associated graph G ( X ) .

Classifying Abelian Groups Through Acyclic Matchings

AbstractThe inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $$\mathbb {Z}^n$$ Z n , subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $$\mathbb {Z}^n$$ Z n . This note aims to classify all abelian groups with respect to the acyclic matching property.

Graded picture invariants and polynomial invariants for mixed tensor superspaces

In this paper we consider the mixed tensor space of a Z2-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, this set turns out to be the set of supertrace monomials, which is analogous to the result of Procesi for the general linear group. A queer supergroup analogue of these results is also obtained.

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

Weak-type estimates of the Bergman projection on a class of singular Reinhardt domains

In this paper, we study the weak-type regularity of the Bergman projection on monomial polyhedra, which is a wide class of bounded singular pseudoconvex Reinhardt domains defined as sublevel sets of holomorphic monomials. We prove that the weak-type estimate of the Bergman projection holds at the upper endpoint of Lp boundedness range, but fails at the lower endpoint. This result generalizes the works of Huo-Wick and Christopherson-Koenig on 2-dimensional Hartogs triangles to a much more general setting, as well as the work of Jing et al. on n-dimensional generalized Hartogs triangle. As a consequence, we also obtain the weak-type regularity on the well-known elementary Reinhardt domains, which has not been considered by others.

Macaulay’s theorem for vector-spread algebras

Let [Formula: see text] be the standard graded polynomial ring, with K a field, and let [Formula: see text], [Formula: see text], be a [Formula: see text]-tuple whose entries are non-negative integers. To a t-spread ideal I in S, we associate a unique [Formula: see text]-vector and we prove that if I is t-spread strongly stable, then there exists a unique t-spread lex ideal which shares the same [Formula: see text]-vector of I via the combinatorics of the t-spread shadows of special sets of monomials of S. Moreover, we characterize the possible [Formula: see text]-vectors of t-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal–Katona. Finally, we prove that among all t-spread strongly stable ideals with the same [Formula: see text]-vector, the t-spread lex ideals have the largest Betti numbers.

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Zigzag strip bundles and crystals II

Zigzag strip bundles are combinatorial models realizing the crystals B(∞) and the highest weight crystals B(λ) over quantum affine algebras Uq(g), and they are closely related with Nakajima monomials and Kashiwara embeddings of specific type. In this paper, we introduce new zigzag strip bundles for Uq(g), which we call reverse zigzag strip bundles in order to distinguish from the previous ones, and we give explicit 1-1 correspondences with the set of Nakajima monomials and the image of Kashiwara embedding which are different from those appeared in the study of the zigzag strip bundles. Another main result is to give different points of view about zigzag strip bundles and reverse zigzag strip bundles. Indeed, the previous 1-1 correspondences between the sets of zigzag strip bundles or reverse zigzag strip bundles, and the sets of Nakajima monomials or images of Kashiwara embeddings were determined based on the number of blocks in the columns of these strip bundles. In our work, we use the number of blocks in the rows of zigzag strip bundles and reverse zigzag strip bundles to provide characterizations of Nakajima monomials and images of Kashiwara embeddings that differ from the previous ones. That is, we give characterizations of Nakajima monomials and Kashiwara embeddings of 4 types. Further, we discuss the connection between zigzag strip bundles and reverse zigzag strip bundles.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm. The source code is made publicly available for reproducibility.

Bergman kernels of monomial polyhedra

The Bergman kernels of monomial polyhedra are explicitly computed. Monomial polyhedra are a class of bounded pseudoconvex Reinhardt domains defined as sublevel sets of Laurent monomials. Their kernels are rational functions and are obtained by an application of Bell's transformation formula.

Generalized bijective maps between G-parking functions, spanning trees, and the Tutte polynomial

We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between G -parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph G . A tree growing sequence determines an algorithm which can be applied to a single function, or to the set P G , q of G -parking functions. When the latter is chosen, the algorithm uses splitting operations – inspired by the recursive definition of the Tutte polynomial – to iteratively break P G , q into disjoint subsets. This results in bijective maps τ and ρ from P G , q to the spanning trees of G and Tutte monomials, respectively. We compare the TGS algorithm to Dhar’s algorithm and the family described by Chebikin and Pylyavskyy in 2005. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.

Tropical Mirror Symmetry in Dimension One

We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [B\"ohm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.

Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information about the GHWs. We prove that the first and second generalized Hamming weights of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated with a binomial ideal related with the code. Moreover, the remaining weights are bounded above by the degrees of the syzygies in the resolution.

Computing generalized hamming weights of binary linear codes via free resolutions

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.

SYMPLECTIC PBW DEGENERATE FLAG VARIETIES; PBW TABLEAUX AND DEFINING EQUATIONS

We define a set of PBW-semistandard tableaux that is in a weight-preserving bijection with the set of monomials corresponding to integral points in the Feigin–Fourier–Littelmann–Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the multihomogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety with respect to the Plücker embedding. These relations consist of type Α degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini’s linear relations.

Toric ideals of weighted oriented graphs

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Data-Driven Sparse Polynomial Chaos Expansion for Models with Dependent Inputs

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram-Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm.

A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets of variables. They can be organized into a cartesian bicategory, which unfortunately fails to be closed for essentially two reasons, which we address here by suitably modifying the model. Firstly, a naive closure is too large to be well-defined, which can be overcome by restricting to polynomials which are finitary. Secondly, the resulting putative closure fails to properly take the 2-categorical structure in account. We advocate here that this can be addressed by considering polynomials in groupoids, instead of sets. For those, the constructions involved into composition have to be performed up to homotopy, which is conveniently handled in the setting of homotopy type theory: we use it here to formally perform the constructions required to build our cartesian bicategory, in Agda. Notably, this requires us introducing an axiomatization in a small universe of the type of finite types, as an appropriate higher inductive type of natural numbers and bijections.

Gorenstein rings generated by strongly stable sets of quadratic monomials

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel.

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ( $$SG$$ ), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. These orthogonal polynomials arise through the Gram–Schmidt orthogonalisation process applied on the set of monomials on $$SG$$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $$L^2$$ , $$L^\infty $$ , and Sobolev norms, and study their asymptotic behavior. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.