arXiv subjects

Lorenzo Robbiano

Publications and source records attributed to Lorenzo Robbiano.

At least 19 recordsLinked to original sources

arXiv2026

Border Bases and Border Basis Schemes

This survey invites the readers on a journey spanning more than twenty years of research through the landscape of border basis schemes. During most of this period, I had the pleasure of working with Martin Kreuzer, and more recently with Le Ngoc Long. Along this journey, one encounters border bases, which are characterized by the remarkable property that their associated multiplication matrices commute pairwise. This property, in turn, provides a natural foundation for defining border basis schemes (BBS). These are beautiful schemes, elegantly defined by simple quadratic equations. However, the number of indeterminates in their coordinate rings can be huge. This necessitates a suitable re-embedding into a polynomial ring with fewer indeterminates. This task is accomplished in a more general setting, and the reward is reaped in the BBS scenario, where the notions of cotangent equivalence and exposed indeterminates play a fundamental role, for instance, in showing that planar Box BBS are affine cells. The center stage is taken by positive $P_0$-algebras and the unimodular matrix problem, which allows us to prove that regular algebras of this kind are free. Finally, we turn our attention to special BBS and interesting subschemes of BBS. Are there no more open problems? Fortunately, many remain, and a selection of these challenges marks not a final destination, but a new horizon, suggesting that this journey may continue in the near future. on the border of the soul bases of unseen photographs, schemes of ancient thoughts, slowly return into poems and theorems L. Robbiano, 2026}

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arXiv2026

Graded Algebras over Polynomial Rings

Given a trivially graded polynomial ring $A=K[a_1,\dots,a_m]$ over a field $K$ and a positively graded polynomial ring $P=A[x_1,\dots,x_k]$, we study graded rings $R=P/I$, where $I$ is a homogeneous ideal in $P$ such that $I\cap A = \{0\}$. The corresponding morphism $Θ: {\rm Spec}(R) \rightarrow {\rm Spec}(A) = \mathbb{A}^m_K$ is used to prove that ${\rm Spec}(R)$ is connected. Then we characterize and compute the following loci in $\mathbb{A}^m_K$: the set ${\rm Sing}_0(Θ)$ of all points such that the corresponding point in the zero section of $Θ$ is singular in ${\rm Spec}(R)$, the set ${\rm Sing}_v(Θ)$ of all points $Γ$ such that the origin of the fiber $F_Γ$ of $Θ$ is singular, and the set ${\rm Sing}_s(Θ)$ of all points $Γ$ such that $\dim({\rm Sing}(F_Γ)) \ge 1$. These results are then used to study MaxDeg border basis schemes, as their coordinate rings are non-negatively graded by the total arrow degree and they have the required structure. In particular, we explicitly determine the singular loci for the $\mathcal{O}$-border basis schemes with $\mathcal{O}=\{1,x,y,z,z^2\}$ and $\mathcal{O} = \{1,x,y,z,yz\}$.

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arXiv2025

Re-Embeddings of Special Border Basis Schemes

Border basis schemes are open subschemes of the Hilbert scheme of $μ$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space $\mathbb{A}^{μν}$. Here we bring together several techniques for re-embedding affine schemes into lower dimensional spaces which we developed in the last years. We study their efficacy for some special types of border basis schemes such as MaxDeg border basis schemes, L-shape and simplicial border basis schemes, as well as planar border basis schemes. A particular care is taken to make these re-embeddings efficiently computable and to check when we actually get an isomorphism with $\mathbb{A}^{nμ}$, i.e., when the border basis scheme is an affine cell.

math.AG↗

arXiv2024

Elimination by Substitution

Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding polynomials such that $f_i = z_i - h_i$ with $h_i \in K[X \setminus Z]$. Here we elaborate on this technique in the case when $P$ is non-negatively graded. The existence of a coherently $Z$-separating tuple is reduced to solving several $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I \longrightarrow K[X \setminus Z] / (I \cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0 \longrightarrow P/I$ as a fibration over an affine space, we show that its fibers allow optimal $Z$-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of $P$ such that additional $Z$-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular.

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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.

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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.

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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.

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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.

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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

Saturations of Subalgebras, SAGBI Bases, and U-invariants

Given a polynomial ring $P$ over a field $K$, an element $g \in P$, and a $K$-subalgebra $S$ of $P$, we deal with the problem of saturating $S$ with respect to $g$, i.e. computing $Sat_g(S) = S[g, g^{-1}]\cap P$. In the general case we describe a procedure/algorithm to compute a set of generators for $Sat_g(S)$ which terminates if and only if it is finitely generated. Then we consider the more interesting case when $S$ is graded. In particular, if $S$ is graded by a positive matrix $W$ and $g$ is an indeterminate, we show that if we choose a term ordering $σ$ of $g$-DegRev type compatible with $W$, then the two operations of computing a $σ$-SAGBI basis of $S$ and saturating $S$ with respect to $g$ commute. This fact opens the doors to nice algorithms for the computation of $Sat_g(S)$. In particular, under special assumptions on the grading one can use the truncation of a $σ$-SAGBI basis and get the desired result. Notably, this technique can be applied to the problem of directly computing some $U$-invariants, classically called semi-invariants, even in the case that $K$ is not the field of complex numbers.

math.AC↗

arXiv2019

Ideals modulo a prime

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the \textit{reduction modulo $p$} of $I$. We first define a new notion of $σ$-good prime for $I$ which does depends on the term ordering $σ$, but not on the given generators of $I$. We relate our notion of $σ$-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo~$p$ from the term ordering, thus letting us show that all but finitely many primes are good for $I$. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

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arXiv2019

Computing and Using Minimal Polynomials

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality.

math.AC↗

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.

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arXiv2019

Small Groebner Fans of Ideals of Points

In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Groebner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Groebner bases, that is, with the same cardinality of their associated Groebner fan.

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.

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arXiv2016

Implicitization of Hypersurfaces

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal via a \textit{truncated, homogeneous} Gröbner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm. Either may be used inside the third algorithm, "RatPar", to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm "ModImplicit", for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

math.AC↗

arXiv2014

Hyperplane Sections, Groebner Bases, and Hough Transforms

The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Gröbner bases pass to the quotient and when they can be lifted. The main difficulty to overcome is the fact that we deal with non-homogeneous ideals. As a by-product we hint at a promising technique for computing implicitization efficiently. In the second part of the paper we deal with families of algebraic schemes and the Hough transforms, in particular we compute their dimension, and show that in some interesting cases it is zero. Then we concentrate on their hyperplane sections. Some results and examples hint at the possibility of reconstructing external and internal surfaces of human organs from the parallel cross-sections obtained by tomography.

math.AC↗

arXiv2012

An Algebraic Approach to Hough Transforms

The main purpose of this paper is to lay the foundations of a general theory which encompasses the features of the classical Hough transform and extend them to general algebraic objects such as affine schemes. The main motivation comes from problems of detection of special shapes in medical and astronomical images. The classical Hough transform has been used mainly to detect simple curves such as lines and circles. We generalize this notion using reduced Groebner bases of flat families of affine schemes. To this end we introduce and develop the theory of Hough regularity. The theory is highly effective and we give some examples computed with CoCoA.

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