Abstract
In [2], a concept of approximate factorization of multivariate polynomial was introduced and two algorithms of approximate factorization were proposed. One algorithm determines the irreducible factors by handling the combinations of roots of the form λ1ϕ i1 +...+λ n ϕ i n , where ϕ1,...,ϕ n are the roots of a given polynomial, λ1,...,λ n are numbers, andi=1, 2,…, and it seems to be practical and important. However, [2] gave only an introductory description of the algorithm and the mathematical as well as computational analysis of the algorithm was postponed. This paper proves completeness of the algorithm by assuming that the numerical coefficients are calculated with an enough accuracy.
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References
B. Buchberger, An algorithmic method in polynomial ideal theory. Multidimensional Systems Theory (ed. Bose, N.K.), Reidel, Dordrecht, 1985.
T. Sasaki, M. Suzuki, M. Kolář and M. Sasaki, Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J. Indust. Appl. Math.,8 (1991), 357–375.
B.L. van der Waerden, Moderne Algebra. Springer Verlag, Berlin, 2nd ed., 1937.
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Work supported in part by Japanese Ministry of Education, Science and Culture under grants #03558008.
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Sasaki, T., Saito, T. & Hilano, T. Analysis of approximate factorization algorithm I. Japan J. Indust. Appl. Math. 9, 351–368 (1992). https://doi.org/10.1007/BF03167271
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DOI: https://doi.org/10.1007/BF03167271