Family Of Identities Research Articles

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

A Deterministic Method for Computing Bertini Type Invariants of Parametric Ideals

A new framework is proposed for computing general numerical invariants, called Bertini type invariants, associated to a family of polynomial ideals depending on deformation parameters. A deterministic algorithm is introduced for computing parameter dependency of Bertini type invariants. As an application, an algorithm for computing Chern–Schwartz–MacPherson classes of a family of singular projective hypersurfaces is obtained.

Ideal Structure of Nica-Toeplitz Algebras

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\mathcal {T}(X)$$\\end{document} of a product system (A, X) over Nn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}^n$$\\end{document}. We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} to A. The main result is a classification of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.

Cohen–Macaulayness of Vertex Splittable Monomial Ideals

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals.

Uniform Symbolic Topologies and Hypersurfaces

We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.

Multigraded algebras and multigraded linear series

AbstractThis paper is devoted to the study of multigraded algebras and multigraded linear series. For an ‐graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standard multigraded algebras, the volume function is not a polynomial in general. However, in the case when the algebra has a decomposable grading, we show that the volume function is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

Fermatean fuzzy semi-prime ideals of ordered semigroups

Abstract Fermatean fuzzy sets (FFSs) are invented to resolve the underlying limitations of intuitionistic fuzzy sets and Pythagorean fuzzy sets. The major goal of this study is to introduce Fermatean fuzzy semi-prime ideals of ordered semigroups. Fermatean fuzzy semi-prime ideals and Fermatean fuzzy prime ideals are introduced. Also, we illustrate some novel concepts to construct Fermatean fuzzy intra-regular and regular ideals. Using the conception of function of the characteristic of ordered semigroups of FFSs, we show certain fundamental facts. Several relations are given for the family of Fermatean fuzzy regular ideals of ordered semigroups.

The semitrivial ideal extension

In this paper, we aim to generalize the ideal duplication defined for numerical semigroups to commutative rings with unity. We introduce the semitrivial ideal extension, a construction that, starting with an ideal of a commutative ring [Formula: see text] with unity and a submodule of a module [Formula: see text] over [Formula: see text], under specific assumptions, produces an ideal of the semitrivial extension [Formula: see text]. Using this tool we characterize the homogeneous prime ideals of a semitrivial extension and we completely describe the family of the maximal ideals. Furthermore, as it was done for the numerical duplication, using the semitrivial ideal extension, we characterize the modules [Formula: see text] such that [Formula: see text] is nearly Gorenstein.

On constructions related to the generalized Taylor complex

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katthän on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a subcomplex of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals.

Equations of the multi-Rees algebra of fattened coordinate subspaces

In this paper, we describe the equations defining the multi-Rees algebra [Formula: see text] ([Formula: see text]), where the ideals are generated by subsets of [Formula: see text]. We also show that a family of binomials whose leading terms are squarefree, form a Gröbner basis for the defining equations with lexicographic order. We show that if we remove binomials that include [Formula: see text]’s, then remaining binomials form a Gröbner basis for the toric ideal associated to the multi-fiber ring. However binomials, including [Formula: see text]’s, in Gröbner basis of defining equations of the multi-Rees algebra are not necessarily defining equations of corresponding symmetric algebra. Despite this fact, we show that this family of ideals is of multi-fiber type.

The spectrum of the Burnside Tambara functor of a cyclic group

We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.

A Volume = Multiplicity formula for 𝑝-families of ideals

In this article, we work with certain families of ideals called p p -families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and F F -signature. For each p p -family of ideals, we attach an Euclidean object called p p -body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of p p -bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for p p -families of m R \mathfrak {m}_{R} -primary ideals in a Noetherian local ring R R .

The saturation number of monomial ideals

Let be the polynomial ring over a field and be the homogeneous maximal ideal of S. For an ideal , let be the minimum number k for which . In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.

Families of Permutations and Ideals of Turing Degrees

Families of Permutations and Ideals of Turing Degrees

On Dualization over Distributive Lattices

Given a partially order set (poset) $P$, and a pair of families of ideals $\mathcal{I}$ and filters $\mathcal{F}$ in $P$ such that each pair $(I,F)\in \mathcal{I}\times\mathcal{F}$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\mathcal{F}$ and does not contain any member of $\mathcal{I}$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $\mathcal{A},\mathcal{B}\subseteq L$ such that no $a\in\mathcal{A}$ is dominated by any $b\in\mathcal{B}$, whether $\mathcal{A}$ and $\mathcal{B}$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $\mathcal{A}$ and $\mathcal{B}$, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.

Using Semicontinuity for Standard Bases Computations

We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K. We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps A={mathbb {Z}} and A=k[t], k any field and t a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for A={mathbb {Z}}, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power.

Convex bodies and asymptotic invariants for powers of monomial ideals

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 monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.

Convex bodies and graded families of monomial ideals

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.

Properties of analogues of Frobenius powers of ideals

Let $$R=\mathbb {K}[X_1, \ldots , X_n ]$$ be a polynomial ring over a field $$\mathbb {K}$$ . We introduce an endomorphism $$\mathcal {F}^{[m]}: R \rightarrow R $$ and denote the image of an ideal I of R via this endomorphism as $$I^{[m]}$$ and call it to be the m-th square power of I. In this article, we study some homological invariants of $$I^{[m]}$$ such as regularity, projective dimension, associated primes, and depth for some families of ideals, e.g., monomial ideals.

Hilbert schemes and y–ification of Khovanov–Rozansky homology

We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $\mathbb{Q}[x_c,y_c]_{c\in \pi_0(L)}$. We conjecture that this invariant restores the missing $Q\leftrightarrow TQ^{-1}$ symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme.