Jinwang Liu
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View article: Equivalent of Multivariate Polynomial Matrix
Equivalent of Multivariate Polynomial Matrix Open
The equivalence of multidimensional systems is closely related to the reduction of multivariate polynomial matrices, with the Smith normal form of matrices playing a key role. So far, the problem of reducing multivariate polynomial matrice…
View article: Further results on equivalence of multivariate polynomial matrices
Further results on equivalence of multivariate polynomial matrices Open
This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial. We first generalized a global-local theorem of Vaserstein. Then we proved these ma…
View article: Bivariate Polynomial Matrix and Smith Form
Bivariate Polynomial Matrix and Smith Form Open
Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This artic…
View article: The algorithm for canonical forms of neural ideals
The algorithm for canonical forms of neural ideals Open
To elucidate the combinatorial architecture of neural codes, the neural ideal $ J_C $, an algebraic object, was introduced. Represented in its canonical form, $ J_C $ provides a succinct characterization of the inherent receptive field arc…
View article: Embedding of Unimodular Row Vectors
Embedding of Unimodular Row Vectors Open
In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonst…
View article: Some improvements for the algorithm of Gröbner bases over dual valuation domain
Some improvements for the algorithm of Gröbner bases over dual valuation domain Open
As a special ring with zero divisors, the dual noetherian valuation domain has attracted much attention from scholars. This article aims at to improve the Buchberger's algorithm over the dual noetherian valuation domain. We present some cr…
View article: Rationalizing Denominators Using Gröbner Bases
Rationalizing Denominators Using Gröbner Bases Open
The problem of rationalizing denominators for two types of fractions is discussed in the paper. By using the theory and algorithms of Gröbner bases, we first introduce a method to rationalize the denominators of fractions with square root …
View article: Some Further Results on the Reduction of Two‐Dimensional Systems
Some Further Results on the Reduction of Two‐Dimensional Systems Open
The reduction of two‐dimensional systems plays an important role in the theory of systems, which is closely associated with the equivalence of the bivariate polynomial matrices. In this paper, the equivalence problems on several classes of…
View article: On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains
On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains Open
In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime fac…
View article: On Serre Reduction of Multidimensional Systems
On Serre Reduction of Multidimensional Systems Open
Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of D …
View article: The Hermite ring conjecture and special linear groups for valuation rings
The Hermite ring conjecture and special linear groups for valuation rings Open
In this paper we prove that the Hermite ring conjecture holds for valuation rings $V$, and the special liner group $SL_n(V[x])$ coincides with the group generated by elementary matrices $E_n(V[x])$ for $n\geq3$. For any arithmetical ring $…
View article: Minor Prime Factorization for <i>n</i>‐D Polynomial Matrices over Arbitrary Coefficient Field
Minor Prime Factorization for <i>n</i>‐D Polynomial Matrices over Arbitrary Coefficient Field Open
In this paper, we investigate two classes of multivariate ( n ‐D) polynomial matrices whose coefficient field is arbitrary and the greatest common divisor of maximal order minors satisfy certain condition. Two tractable criterions are pres…
View article: Generalized Serre Problem over Elementary Divisor Rings
Generalized Serre Problem over Elementary Divisor Rings Open
Matrix factorization has been widely investigated in the past years due to its fundamental importance in several areas of engineering. This paper investigates completion and zero prime factorization of matrices over elementary divisor ring…