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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Mathematische Zeitschrift 148 (1976), S. 107-118 
    ISSN: 1432-1823
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    Library Location Call Number Volume/Issue/Year Availability
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Applicable algebra in engineering, communication and computing 4 (1993), S. 217-230 
    ISSN: 1432-0622
    Keywords: Algebraic variety decomposition ; Gröbner bases ; Systems of nonlinear equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics , Technology
    Notes: Abstract This paper deals with systems ofm polynomial equations inn unknown, which have only finitely many solutions. A method is presented which decomposes the solution set into finitely many subsets, each of them given by a system of type $$f_1 \left( {x_1 } \right) = 0,f_2 \left( {x_1 ,x_2 } \right) = 0, \ldots ,f_n \left( {x_1 , \ldots ,x_n } \right) = 0$$ . The main tools for the decomposition are from ideal theory and use symbolical manipulations. For the ideal generated by the polynomials which describe the solution set, a lexicographical Gröbner basis is required. A particular element of this basis allows the decomposition of the solution set. By a recursive application of these decomposition techniques the triangular subsystems are finally obtained. The algorithm gives even for non-finite solution sets often also usable decompositions.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 70 (1995), S. 311-329 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 12D10, 26D10, 30C15, 65H10, 65H15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The eigenproblem method calculates the solutions of systems of polynomial equations $ f_1(x_1, \ldots , x_s)=0,\ldots,f_m(x_1, \ldots , x_s)=0$ . It consists in fixing a suitable polynomial $ f $ and in considering the matrix $ A_f $ corresponding to the mapping $ [p] \mapsto [f\cdot p] $ where the equivalence classes are modulo the ideal generated by $ f_1, \ldots , f_m.$ The eigenspaces contain vectors, from which all solutions of the system can be read off. This access was investigated in [1] and [16] mainly for the case that $ A_f is nonderogatory. In the present paper, we study the case where $ f_1, \ldots , f_m $ have multiple zeros in common. We establish a kind of Jordan decomposition of $ A_f $ reflecting the multiplicity structure, and describe the conditions under which $ A_f $ is nonderogatory. The algorithmic analysis of the eigenproblem in the general case is indicated.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Advances in computational mathematics 12 (2000), S. 335-362 
    ISSN: 1572-9044
    Keywords: ideal bases ; Gröbner bases ; multivariate polynomials ; interpolation ; systems of polynomial equations ; 65D05 ; 65H10 ; 13P10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.
    Type of Medium: Electronic Resource
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  • 5
    Publication Date: 2018-12-06
    Description: Gröbner bases are the main tool for solving systems of algebraic equations and some other problems in connection with polynomial ideals using Computer Algebra Systems. The procedure for the computation of Gröbner bases in REDUCE 3.3 has been modified in order to solve more complicated algebraic systems of equations by some general improvements and by some tools based on the specific resources of the CRAY X-MP. We present this modification and illustrate it by examples.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 6
    Publication Date: 2018-12-06
    Description: The paper presents a new application of computer algebra to the treatment of steady states of reaction systems. The method is based on the Buchberger algorithm. This algorithm was modified such that it can exploit the special structure of the equations derived from reaction systems, so even large systems can be handled. In contrast to numerical approximation techniques, the algebraic solution gives a complete and definite overview of the solution space and it is even applicable when parameter values are unknown or undetermined. The algorithm, its adaptation to the problem class and its application to selected examples are presented.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 7
    Publication Date: 2014-02-26
    Description: In this paper we consider the problem of reconstructing a multivariate rational function, when only its values at sufficiently many points are known. We use for the reconstruction of bivariate rational functions a bivariate rational interpolation operator investigated by Siemaszko [7] and a new one, compare both by examples in a Computer Algebra system, and present their multivariate generalizations. {\bf Keywords:} Multivariate rational interpolation, reconstruction, symbolic computation.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 8
    Publication Date: 2014-02-26
    Description: This paper deals with systems of $m$ polynomial equations in $n$ unknown, which have only finitely many solutions. A method is presented which decomposes the solution set into finitely many subsets, each of them given by a system of type \begin{displaymath} f_1(x_1)=0, f_2(x_1,x_2)=0,...,f_n(x_1,...,x_n)=0. \end{displaymath} The main tools for the decomposition are from ideal theory and use symbolical manipulations. For the ideal generated by the polynomials which describe the solution set, a lexicographical Gröbner basis is required. A particular element of this basis allows the decomposition of the solution set. A recursive application of these decomposition techniques gives finally the triangular subsystems. The algorithm gives even for non-finite solution sets often also usable decompositions. {\bf Keywords:} Algebraic variety decomposition, Gröbner bases, systems of nonlinear equations.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/x-tar
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