arXiv subjects

Thieu N. Vo

Publications and source records attributed to Thieu N. Vo.

24 records · Page 2Linked to original sources

arXiv2021

On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.

cs.SC↗

arXiv2021

Testing isomorphism of complex and real Lie algebras

In this paper, we give algorithms for determining the existence of isomorphism between two finite-dimensional Lie algebras and compute such an isomorphism in the affirrmative case. We also provide algorithms for determining algebraic relations of parameters in order to decide whether two parameterized Lie algebras are isomorphic. All of the considered Lie algebras are considered over a field $\F$, where $\F=\C$ or $\F=\R$. Several illustrative examples are given to show the applicability and the effectiveness of the proposed algorithms.

math.RA↗

arXiv2020

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

This paper concerns the problem of classifying finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. On the one hand, we present an effective method to classify all $(n+1)$-dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of $n$-dimensional nilpotent Lie algebras is given. On the other hand, the problem of classifying all $(n+2)$-dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we provide a method to classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation.

math.RA↗

arXiv2019

Rational Solutions of First-Order Algebraic Ordinary Difference Equations

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees of its rational solutions, and thus derive a complete algorithm for computing corresponding rational solutions.

cs.SC↗

arXiv2018

Complexity of Triangular Representations of Algebraic Sets

Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szanto. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

math.AG↗

arXiv2018

Rational Solutions of High-Order Algebraic Ordinary Differential Equations

We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this condition is called \emph{noncritical}. We prove that usual low order classes of AODEs are noncritical. For rational solutions, we determine a class of AODEs, which are called \emph{maximally comparable}, such that the poles of their rational solutions are recognizable from their coefficients. This generalizes a fact from linear AODEs, that the poles of their rational solutions are the zeros of the corresponding highest coefficient. An algorithm for determining all rational solutions, if there is any, of certain maximally comparable AODEs, which covers $78.54\%$ AODEs from a standard differential equations collection by Kamke, is presented.

cs.SC↗