Narrow Results

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However the usual standard basis algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it, which reduces the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In ${\mathbb{P}}^2$, we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets $X$ and $X^{\prime }$ of collinear points, we get conditions for $X\star X^{\prime }$ to be either a collinear finite set of points or a grid of $\left|X\right|\left|X^{\prime }\right|$ points. In ${\mathbb{P}}^3,$ under suitable conditions (which we prove to be generic), we show that $X\star X^{\prime }$ consists of $\left|X\right|\left|X^{\prime }\right|$ points on the two different rulings of a nondegenerate quadric and we compute its Hilbert function in the case $\left|X\right|=\left|X^{\prime }\right|.$

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

Given a 0-dimensional scheme $𝕏$ in a projective space ${\mathbb{P}}_K^n$ over a field $K$, we characterize the Cayley–Bacharach property (CBP) of $𝕏$ in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley–Bacharach schemes by Dedekind’s formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

In this paper, we recall the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth, in particular, a new generalization of Gotzmann’s Persistence Theorem, the presence of a GCD for a truncation of the ideal, and applications to saturated ideals.

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $I^{(s)}/I^s$ for any fixed $s$. Also for those class of graphs, we compute the Hilbert function of the module $I^{(s)}/I^s$ for all $s.$

We find a necessary and sufficient condition for the Hilbert function to support the ith linear syzygy using type vectors and also construct an Artinian level algebra whose minimal free resolution supports the nth linear syzygy using both a mapping cone construction and coordinate points in ℙⁿ.

Let X be a zero-dimensional scheme in ℙ¹ × ℙ¹. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

In a Cohen-Macaulay local ring $(A,\mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequalityi ${\ell }_A(A/I)\ge {\mathrm{e}}_0(I)-{\mathrm{e}}_1(I)+({\mathrm{e}}_2(I)+{\ell }_A(I^2/QI))/2$, where ${\mathrm{e}}_i(I)$ denotes the $i$th Hilbert coefficient and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.