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  • 1
    Electronic Resource
    Electronic Resource
    H-bases for polynomial interpolation and system solving (2000)
    Springer
    Advances in computational mathematics 12 (2000), S. 335-362 
    Details
    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
    URL: http://dx.doi.org/10.1023/A:1018937723499
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  • 2
    Electronic Resource
    Electronic Resource
    Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems (1995)
    Springer
    Numerische Mathematik 70 (1995), S. 311-329 
    Details
    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
    URL: http://dx.doi.org/10.1007/s002110050122
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  • 3
    Electronic Resource
    Electronic Resource
    Mehrdimensionale Hermite-Interpolation und numerische Integration (1976)
    Springer
    Mathematische Zeitschrift 148 (1976), S. 107-118 
    Details
    ISSN: 1432-1823
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01214700
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  • 4
    Electronic Resource
    Electronic Resource
    On decomposing systems of polynomial equations with finitely many solutions (1993)
    Springer
    Applicable algebra in engineering, communication and computing 4 (1993), S. 217-230 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01200146
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