arXiv subjects

Le Ngoc Long

Publications and source records attributed to Le Ngoc Long.

16 recordsLinked to original sources

arXiv2025

The Canonical Exact Sequence of Differential Modules for 0-Dimensional Schemes

Given a 0-dimensional scheme $\X$ in $\mathbb{P}^n_K$ over a perfect field $K$, we examine the second differential power of its homogeneous vanishing ideal. This enables us to establish the canonical exact sequence for the associated Kähler differential module. We also provide a formula for the Hilbert polynomial of Kähler differential modules when $\X$ is either a fat point scheme or a 0-dimensional locally monomial Gorenstein scheme.

math.AG↗

arXiv2024

Efficiently Checking Separating Indeterminates

In this paper we continue the development of a new technique for computing elimination ideals by substitution which has been called $Z$-separating re-embeddings. Given an ideal $I$ in the polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, this method searches for tuples $Z=(z_1,\dots,z_s)$ of indeterminates with the property that $I$ contains polynomials of the form $f_i = z_i - h_i$ for $i=1,\dots,s$ such that no term in $h_i$ is divisible by an indeterminate in $Z$. As there are frequently many candidate tuples $Z$, the task addressed by this paper is to efficiently check whether a given tuple $Z$ has this property. We construct fast algorithms which check whether the vector space spanned by the generators of $I$ or a somewhat enlarged vector space contain the desired polynomials $f_i$. We also extend these algorithms to Boolean polynomials and apply them to cryptoanalyse round reduced versions of the AES cryptosystem faster.

math.AC↗

arXiv2024

Some line and conic arrangements and their Waldschmidt constants

We study the Waldschmidt constant of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set $\mathbb{X}$ of $n$ points where at least $n-3$ points among them lie on a line is either equal to $1, \frac{2n-3}{n-1}, 2, \frac{16}{7}, \frac{7}{3}, \frac{17}{7},$ or $\frac{5}{2}$. Together with the Hilbert polynomials, this gives a complete geometric characterization for $\mathbb{X}$. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by $\frac{5}{2}$. Under this condition, we describe all configurations of $n$ points with $n-1$ points among them lying on an irreducible conic, and we also study some specific configurations of $9$ points.

math.CO↗

arXiv2024

Re-embeddings of Affine Algebras Via Gröbner Fans of Linear Ideals

Given an affine algebra $R=K[x_1,\dots,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P'/I'$ such that $P'=K[y_1,\dots,y_m]$ has fewer indeterminates. For cases when the number of indeterminates $n$ is large and Gröbner basis computations are infeasible, we have previously introduced the method of $Z$-separating re-embeddings. This method tries to detect polynomials of a special shape in $I$ which allow us to eliminate the indeterminates in the tuple $Z$ by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples $Z$ can be found using the Gröbner fan of the linear part of $I$. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the case of binomial linear ideals using a cotangent equivalence relation. Finally, we apply the improved technique in the case of the defining ideals of border basis schemes.

math.AC↗

arXiv2023

Optimal Re-Embeddings of Border Basis Schemes

Border basis schemes are open subschemes of Hilbert schemes parametrizing 0-dimensional subschemes of $\mathbb{P}^n$ of given length. They yield open coverings and are easy to describe and to compute with. Our topic is to find re-embeddings of border basis schemes into affine spaces of minimal dimension. Given $P = K[X] = K[x_1,\dots,x_n]$, an ideal $I\subseteq \langle X \rangle$, and a tuple $Z$ of indeterminates, in previous papers the authors developed techniques for computing $Z$-separating re-embeddings of $I$, i.e., of isomorphisms $Φ: P/I \rightarrow K[X\setminus Z] / (I\cap K[X\setminus Z])$. Here these general techniques are developed further and improved by constructing a new algorithm for checking candidate tuples $Z$ and by using the Gröbner fan of the linear part of $I$ advantageously. Then we apply this to the ideals defining border basis schemes $\mathbb{B}_{\mathcal{O}}$, where $\mathcal{O}$ is an order ideal of terms, and to their natural generating polynomials. The fact that these ideals are homogeneous w.r.t. the arrow grading allows us to look for suitable tuples $Z$ more systematically. Using the equivalence of indeterminates modulo the square of the maximal ideal, we compute the Gröbner fan of the linear part of the ideal quickly and determine which indeterminates should be in $Z$ when we are looking for optimal re-embeddings. Specific applications include re-embeddings of border basis schemes where $\mathcal{O}\subseteq K[x,y]$ and where $\mathcal{O}$ consists of all terms up to some degree.

math.AG↗

arXiv2022

Restricted Gröbner fans and re-embeddings of affine algebras

In this paper we continue the study of good re-embeddings of affine K-algebras started in [KLR]. The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where X=(x_1,...,x_n), and K-algebras having fewer generators. These projections are induced by particular tuples of indeterminates Z and by term orderings $σ$ which realize Z as leading terms of a tuple F of polynomials in I. In order to efficiently find such tuples, we provide two major new tools: an algorithm which reduces the check whether a given tuple F is Z-separating to an LP feasibility problem, and an isomorphism between the part of the Gröbner fan of I consisting of marked reduced Gröbner bases which contain a Z-separating tuple and the Gröbner fan of the intersection of I and K[X\Z]. We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.

math.AC↗

arXiv2022

The Kähler different of a 0-dimensional scheme

Given a 0-dimensional scheme $\mathbb{X}$ in the projective $n$-space $\mathbb{P}^n_K$ over a field $K$, we are interested in studying the Kähler different of $\mathbb{X}$ and its applications. Using the Kähler different, we characterize the generic position and Cayley-Bacharach properties of $\mathbb{X}$ in several certain cases. When $\mathbb{X}$ is in generic position, we prove a generalized version of the Apéry-Gorenstein-Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the Kähler different and the Cayley-Bacharach property.

math.AC↗

arXiv2021

Cotangent spaces and separating re-embeddings

Given an affine algebra $R=P/I$, where $P=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme ${\rm Spec}(R)$, i.e., presentations $R \cong P'/I'$ such that $P'$ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i= z_i - g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of $I$. The dimension of the cotangent space of $R$ at a $K$-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of $R$ and found an optimal re-embedding.

math.AC↗

arXiv2020

Computing subschemes of the border basis scheme

A good way of parametrizing 0-dimensional schemes in an affine space $\mathbb{A}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $μ$, they provide an open covering of the Hilbert scheme ${\rm Hilb}^μ(\mathbb{A}^n_K)$ and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent 0-dimensional $K$-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley-Bacharach, and strict Cayley-Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by non-trivial, concrete examples.

math.AC↗

arXiv2020

Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Given a fat point scheme $\mathbb{W}=m_1P_1+\cdots+m_sP_s$ in the projective $n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules of Kähler differential $k$-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $\mathbb{W}$ when $k\in\{1,\dots, n+1\}$. In this paper we determine the value of its Hilbert polynomial explicitly for the case $k=n+1$, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme $\mathbb{Y} = (m_1-1)P_1 + \cdots + (m_s-1)P_s$. For $n=2$, this allows us to determine the Hilbert polynomials of the modules of Kähler differential $k$-forms for $k=1,2,3$, and to produce a sharp bound for the regularity index for $k=2$.

math.AG↗

arXiv2019

Algorithms for Checking Zero-Dimensional Complete Intersections

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe's characterisation of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

math.AC↗

arXiv2019

An application of Liaison theory to zero-dimensional schemes

Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property.

math.AC↗

arXiv2018

On the Cayley-Bacharach Property

The Cayley-Bacharach property, which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the Cayley-Bacharach property directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the Cayley-Bacharach property and the symmetry of their affine Hilbert function, as well as by the strict Cayley-Bacharach property and the last difference of their affine Hilbert function.

math.AC↗

arXiv2018

Kaehler differentials for fat point schemes in P^1xP^1

Let $X$ be a set of $K$-rational points in $P^1 \times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $Ω^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 \times P^1$ having the Cayley-Bacharach property.

math.AG↗

arXiv2017

On the Dedekind different of a Cayley-Bacharach scheme

Given a 0-dimensional scheme $\mathbb{X}$ in a projective space $\mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $\mathbb{X}$ 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.

math.AG↗

arXiv2017

Kähler differential algebras for 0-dimensional schemes

Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a field $K$, we study the Kähler differential algebra $Ω_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules $Ω^m_{R/K}$ of Kähler differential $m$-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of $\mathbb{P}^1$, fat point schemes, and subschemes of $\mathbb{P}^2$ supported on a conic.

math.AC↗