Serre’s conjecture, stating that every finitely generated projective module over a polynomial ring is free, was proven by Quillen and Suslin independently in 1976. An equivalent form of the Quillen-Suslin theorem says, “Every unimodular row over a polynomial ring can be completed to a unimodular matrix.” In this paper, we generalize the Quillen-Suslin theorem to the parametric case and present an algorithm to construct the unimodular completion matrix system for any polynomial vector with parameters. Specifically, we first determine the conditions on the parameters under which the vector is unimodular using the comprehensive Gröbner system. Furthermore, we use a constructive method to find a finite partition of the parameter space such that, for each branch, the vector under specializations can be completed into a unimodular matrix in the same form. Since the method is constructive, we present an explicit algorithm to construct the unimodular completion matrix system for any polynomial vector with parameters. The correctness and termination of the algorithm have been proven, and an example is provided to demonstrate how the algorithm works.

A new algorithm for computing the parametric greatest common right divisor (GCRD) of a set of parametric Ore polynomials is presented in this paper. The algorithm is based on Gröbner bases for modules. Inspired by the resultant theory in Ore polynomial rings, the Sylvester matrix is defined for a set of Ore polynomials. In the case of non-parametric polynomials, the GCRD of Ore polynomials can be obtained by computing the row echelon form of the Sylvester matrix. For the parametric case, the parametric Sylvester matrix is also defined in the paper. Based on this, under the assumption that the specializations commute with the conjugate operator and derivation in the Ore polynomial ring, the parametric GCRD of parametric Ore polynomials can be obtained by computing the Gröbner basis for the module generated by rows of the parametric Sylvester matrix. As a consequence, the algorithm for computing the parametric GCRD is presented in detail and has been implemented in the computer algebra system Singular.

Abstract Recent dramatic increases in the ability to generate, collect, and use datasets have inspired numerous academic and policy discussions regarding the emerging field of human data interaction (HDI). Given the challenges in interacting with open government data (OGD) and the existing research gap in this field, our study intends to explore HDI in the OGD domain and investigate ways HDI can further contribute to OGD promotion. Building upon two existing behavioral models, we proposed an initial conceptual model for OGD interaction, then using this model, conducted two studies to empirically examine users' behaviors when interacting with OGD. Ultimately, we refined this model for OGD interaction and invited three experts to validate it to enhance its understandability, comprehensiveness, and reasonableness. This comprehensive model for human OGD interaction will contribute to the theoretical work of the HDI field as well as the practical design of OGD platforms and data literacy education.

This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.

An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.

The rapid development of open government data and the increasing attention on data use/reuse have stimulated many studies on data-related issues. The extant studies show that even though OGD portals have been rapidly developed in this age of data, there are still various challenges when users interact with data. Therefore, this dissertation attempts to identify the contextualized user challenges, understand user behaviors when interacting with OGD, and ultimately, propose solutions that can assist users in using OGD. Furthermore, aiming to ameliorate the issues that data is hard to find and to be understood, a mixed-method design will be employed, including a transaction log analysis, a content analysis, and semistructured interviews. Also, the research sites accessed are two local-level OGD portals: OpenDataPhilly and Western Pennsylvania Regional Data Center. Local-level portals are closely connected with local communities and neighborhoods, which have a direct impact on a citizen's daily life. The results of this dissertation are expected to contribute to the fields of Human Information/Data Interaction, Human Computer Interaction and OGD Use.

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present a new criterion for the existence of matrix factorizations for a class of multivariate polynomial matrices, and prove that these matrix factorizations are unique. Based on this new criterion and the constructive proof process, we give an algorithm to compute a matrix factorization of a multivariate polynomial matrix. After that, we put forward a sufficient and necessary condition for the equivalence of square polynomial matrices: a square polynomial matrix is equivalent to a diagonal triangle if it satisfies the condition. An illustrative example is given to show the effectiveness of the matrix equivalence theorem.

An extended greatest common divisor (GCD) algorithm for parametric univariate polynomials is presented in this paper. This algorithm computes not only the GCD of parametric univariate polynomials in each constructible set but also the corresponding representation coefficients (or multipliers) for the GCD expressed as a linear combination of these parametric univariate polynomials. The key idea of our algorithm is that for non-parametric case the GCD of arbitrary finite number of univariate polynomials can be obtained by computing the minimal Gröbner basis of the ideal generated by those polynomials. But instead of computing the Gröbner basis of the ideal generated by those polynomials directly, we construct a special module by adding the unit vectors which can record the representation coefficients, then obtain the GCD and representation coefficients by computing a Gröbner basis of the module. This method can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCD algorithm for parametric univariate polynomials. More importantly, we apply the proposed extended GCD algorithm to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.

The importance of open government data is often associated with increased public trust, civic engagement, and accountable administrations. While there is a myriad of benefits, the existing literature suggests that many open government datasets lack accessibility and usability for diverse users. This study seeks to explore what contextual information users require when they access these datasets. Using mixed methods, we aim to discover the challenges of accessing data, and the necessary contextual information needed by the users to overcome these challenges. As the outcome of this study, we propose a framework called "Data Guides", which is composed of the identified important contextual information. In future work, we will test the effectiveness of the Data Guide in aiding users' accessing and understanding open government data.

A new algorithm, which combines the GVW algorithm with the Mora normal form algorithm, is presented to compute the standard bases of ideals in a local ring. Since term orders in local ring are not well-orderings, there may not be a minimal signature in an infinite set, and we can not extend the GVW algorithm from a polynomial ring to a local ring directly. Nevertheless, when given an anti-graded order in R and a term-over-position order in Rm that are compatible, we can construct a special set such that it has a minimal signature, where R , Rm are a local ring and a R -module, respectively. That is, for any given polynomial v0 ın R, the set consisting of signatures of pairs (u,v)ın Rm x R has a minimal element, where the leading power products of v and v0 are equal. In this case, we prove a cover theorem in R , and use three criteria (syzygy criterion, signature criterion and rewrite criterion) to discard useless J-pairs without any reductions. Mora normal form algorithm is also extended to do regular top-reductions in Rm x R, and the correctness and termination of the algorithm are proved. The proposed algorithm has been implemented in the computer algebra system Maple, and experiment results show that most of J-pairs can be discarded by three criteria in the examples.