arXiv subjects

Sonia L. Rueda

Publications and source records attributed to Sonia L. Rueda.

At least 19 recordsLinked to original sources

arXiv2026

Gröbner bases of Burchnall-Chaundy ideals for ordinary differential operators

The correspondence between commutative rings of ordinary differential operators (ODOs) and algebraic curves was established by Burchnall and Chaundy, Krichever and Mumford, among many others. To make this correspondence computationally effective, in this paper we aim to compute the defining ideals of spectral curves, Burchnall-Chaundy (BC) ideals. We provide an algorithm to compute a Gröbner basis of a BC ideal. The point of departure is the computation of the finite set of generators of a maximal commutative ring of ODOs, which was implemented by the authors in the package dalgebra of SageMath. The algorithm to compute BC ideals has been also implemented in dalgebra. The differential Galois theory of the corresponding spectral problems, linear differential equations with parameters, would benefit from the computation on this prime ideal, generated by constant coefficient polynomials. In particular, we prove the primality of the differential ideal generated by a BC ideal, after extending the coefficient field. This is a fundamental result to develop Picard-Vessiot theory for spectral problems.

math.AC↗

arXiv2025

Computing almost commuting bases of ODOs and Gelfand-Dickey hierarchies

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weighted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence, the computation of the equations of the GD hierarchies is achieved without using pseudo-differential operators. A new package in SageMath called \texttt{dalgebra} has been designed to perform symbolic calculations in differential domains. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.

math.AC↗

arXiv2025

Effective computation of centralizers of ODOs

This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of polynomials $C [L]$, with coefficients in the field of constants $C$ of $L$. We give an algorithm to compute a basis of $Z (L)$ as a $C [L]$-module. Our approach combines results by K. Goodearl in 1985 with solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchy, which after substituting the coefficients of $L$ become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ belong to a differential algebraic extension $K$ of $C$. In addition, by considering parametric coefficients we develop an algorithm to generate families of ODOs with non trivial centralizer, in particular algebro-geometric, whose coefficients are solutions in $K$ of systems of the stationary GD hierarchy.

math.RA↗

arXiv2023

Computing defining ideals of space spectral curves for algebro-geometric third order ODOs

Commuting pairs of ordinary differential operators (ODOs) have been related to plane algebraic curves since the work of Burchnall and Chaundy a century ago. We introduce now the concept of Burchnall-Chaundy (BC) ideal of a commuting pair, as the ideal of all constant coefficient bivariate polynomials satisfied by the pair. We prove this prime ideal to be equal to the radical of a differential elimination ideal and the defining ideal of a plane algebraic curve, the spectral curve of a commuting pair. The ODOs of this work have coefficients in an arbitrary differential field with field of constants algebraically closed and of characteristic zero. Motivated by the extension of the recently introduced Picard-Vessiot theory for spectral problems $L(y)=λy$, where $λ$ is an algebraic parameter, we also define the BC ideal of an algebro-geometric third order operator $L$. This allows a constructive proof of a famous theorem by I. Schur, establishing an isomorphism between the centralizer of $L$ and the coordinate ring of a space algebraic curve, that we define as the spectral curve of $L$ and whose defining ideal is the BC ideal of $L$. We provide the first explicit example of a non-planar spectral curve. We compute a set of generators of the defining ideal of this curve by means of differential resultants and define a new coefficient field determined by the spectral curve, to effectively compute an intrinsic right factor of $L-λ$.

math.AG↗

arXiv2023

Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory

Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute there exists a polynomial $f(λ,μ)$ with complex coefficients such that $f(P,Q)=0$, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Matrices have entries in a differential field $K$, whose field of constants $C$ is algebraically closed and of zero characteristic. We restrict to the case of order one operators $P$, with invertible leading coefficient. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem $PY=λY \ , \ QY=μY$ to have a solution. Techniques from differential algebra and Picard-Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs $C[P,Q]$ and a finite product of rings of irreducible algebraic curves.

math.AG↗

arXiv2021

Spectral Picard-Vessiot fields for Algebro-geometric Schrödinger operators

This work is a galoisian study of the spectral problem $LΨ=λΨ$, for algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose field of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $λ$ an algebraic variable over $C$, forcing the consideration of a new field of coefficients for $L-λ$, whose field of constants is the field $C(Γ)$ of the spectral curve $Γ$. Since $C(Γ)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over $Γ$, called "Spectral Picard-Vessiot field" of $L-λ$. An existence theorem is proved using differential algebra, allowing to recover classical Picard-Vessiot theory for each $ λ= λ_0 $. For rational spectral curves, the appropriate algebraic setting is established to solve $LΨ=λΨ$ analitically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

math.SP↗

arXiv2021

Factoring Third Order Ordinary Differential Operators over Spectral Curves

We consider the classical factorization problem of a third order ordinary differential operator $L-λ$, for a spectral parameter $λ$. It is assumed that $L$ is an algebro-geometric operator, that it has a nontrivial centralizer, which can be seen as the affine ring of curve, the famous "spectral curve" $Γ$. In this work we explicitly describe the ring structure of the centralizer of $L$ and, as a consequence, we prove that $Γ$ is a space curve. In this context, the first computed example of a non-planar spectral curve arises, for an operator of this type. Based on the structure of the centralizer, we give a symbolic algorithm, using differential subresultants, to factor $L-λ_0$ for all but a finite number of points $P=(λ_0 , μ_0 , γ_0)$ of the spectral curve .

math.AG↗

arXiv2020

Commuting Ordinary Differential Operators and the Dixmier Test

The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.

math.AG↗

arXiv2019

Factorization of KdV Schrödinger operators using differential subresultants

We address the classical factorization problem of a one dimensional Schrödinger operator $-\partial^2+u-λ$, for a stationary potential $u$ of the KdV hierarchy but, in this occasion, a "parameter" $λ$. Inspired by the more effective approach of Gesztesy and Holden to the "direct" spectral problem, we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make our method fully effective, we design a symbolic algorithm to compute the integration constants of the KdV hierarchy, in the case of KdV potentials that become rational under a Hamiltonian change of variable. Explicit computations are carried for Schrödinger operators with solitonic potentials.

nlin.SI↗

arXiv2014

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from $\mathcal{P}$ through derivations and multiplications by Laurent monomials. To start, through derivations, a system $ps(\mathcal{P})$ of $L$ polynomials in $L-1$ algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in $ps(\mathcal{P})$, to obtain polynomials in the differential elimination ideal generated by $\mathcal{P}$. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

math.AP↗

arXiv2013

Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves

In this paper we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions to the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.

math.AG↗

arXiv2012

Approximate Parametrization of Space Algebraic Curves

Given a non-rational real space curve and a tolerance $ε>0$, we present an algorithm to approximately parametrize the curve. The algorithm checks whether a planar projection of the space curve is $ε$-rational and, in the affirmative case, generates a planar parametrization that is lifted to an space parametrization. This output rational space curve is of the same degree as the input curve, both have the same structure at infinity, and the Hausdorff distance between them is always finite.

math.AG↗

arXiv2012

Linear sparse differential resultant formulas

Let $\cP$ be a system of $n$ linear nonhomogeneous ordinary differential polynomials in a set $U$ of $n-1$ differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in $U$ from $\cP$. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in $\cP$, or in a linear perturbation $\cP_{\varepsilon}$ of $\cP$. In particular, the formula $\dfres(\cP)$ is the determinant of a matrix $\cM(\cP)$ having no zero columns if the system $\cP$ is "super essential". As an application, if the system $\frak{P}$ is sparse generic, such formulas can be used to compute the differential resultant $\dres(\frak{P})$ introduced by Li, Gao and Yuan in (Proceedings of the ISSAC'2011).

math.CA↗

arXiv2010

Parametrization of $ε$-rational curves: error analysis

In [Computer Aided Geometric Design 27 (2010), 212-231] the authors present an algorithm to parametrize approximately $ε$-rational curves, and they show in 2 examples that the Hausdorff distance, w.r.t. to the Euclidean distance, between the input and output curves is small. In this paper, we analyze this distance for a whole family of curves randomly generated and we automatize the strategy used in [Computer Aided Geometric Design 27 (2010), 212-231]. We find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.

math.AG↗

arXiv2010

A perturbed differential resultant based implicitization algorithm for linear DPPEs

Let $\bbK$ be an ordinary differential field with derivation $\partial$. Let $\cP$ be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal $\id$. Given a nonzero linear differential polynomial $A$ in $\id$ we give necessary and sufficient conditions on $A$ for $\cP$ to be $n-1$ dimensional. We prove the existence of a linear perturbation $\cP_ϕ$ of $\cP$ so that the linear complete differential resultant $\dcres_ϕ$ associated to $\cP_ϕ$ is nonzero. A nonzero linear differential polynomial in $\id$ is obtained from the lowest degree term of $\dcres_ϕ$ and used to provide an implicitization algorithm for $\cP$.

math.CA↗

arXiv2009

Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $ε>0$ and an $ε$-irreducible algebraic affine plane curve $\mathcal C$ of proper degree $d$, we introduce the notion of $ε$-rationality, and we provide an algorithm to parametrize approximately affine $ε$-rational plane curves, without exact singularities at infinity, by means of linear systems of $(d-2)$-degree curves. The algorithm outputs a rational parametrization of a rational curve $\bar{\mathcal C}$ of degree at most $d$ which has the same points at infinity as $\mathcal C$. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that $\bar{\mathcal C}$ and $\mathcal C$ are close in practice.

math.AG↗

arXiv2008

Linear Complete Differential Resultants and the Implicitization of Linear DPPEs

The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters. We give necessary conditions to ensure properness of the system of differential polynomial parametric equations.

math.CA↗

arXiv2006

On the computation of graded components of Laurent polynomial rings

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly closed field of characteristic 0. We define the multigrading of $S$ by an arbitrary finitely generated abelian group $A$. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of $S$ using polytopes. We give an algorithm to check whether the graded components of $S$ are finite dimensional. Regardless of the dimension, we determine a finite set of generators of each graded component as a module over the component of homogeneous polynomials of degree 0.

math.AC↗