Abstract
In this paper, the so-called invertibility is introduced for rational univariate representations, and a characterization of the invertibility is given. It is shown that the rational univariate representations, obtained by both Rouillier’s approach and Wu’s method, are invertible. Moreover, the ideal created by a given rational univariate representation is defined. Some results on invertible rational univariate representations and created ideals are established. Based on these results, a new approach is presented for decomposing the radical of a zero-dimensional polynomial ideal into an intersection of maximal ideals.
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References
Rouillier F, Solving zero-dimensional systems through the rational univariate representation, AAECC, 1995, 9: 433–461.
Zeng G X and Xiao S J, Computing the rational univariate representations for zero-dimensional systems by Wu’s method, Sci. Sin. Math., 2010, 40(10): 999–1016 (in Chinese).
Xiao S J and Zeng G X, Algorithms for computing the global infimum and minimum of a polynomial function, Sci. Sin. Math., 2011, 41(9): 759–788 (in Chinese).
Xiao S J and Zeng G X, Solving the equality-constrained minimization problem of polynomial functions, J. Global Optimization, 2019, 75: 683–733.
Wu W T, Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving, Science Press/Kluwer Academic Publishers, Beijing/Dordrecht-Boston-London, 2000.
Wang D K, The software wsolve: A Maple package for solving system of polynomial equations, available at http://www.mmrc.iss.ac.cn/dwang/wsolve.html.
Nagata M, Field Theory, Marcel Dekker, Inc., New York, 1977.
Becker T, Weispfenning V, and Kredel H, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer-Verlag, New York-Berlin-Heidelberg, 1993.
Atiyah M F and MacDonald J G, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.
Jacobson N, Basic Algebra I, 2nd Edition, W. H. Freeman and Company, New York, 1985.
Mishra B, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer-Verlag, New York-Berlin-Heidelberg, 1993.
Gonzales-Vega L, Rouillier F, and Roy M F, Symbolic recipes for polynomial system solving, Eds. by Cohen A M, Cuypers H, and Sterk H, Some Tapas of Computer Algebra, Springer-Verlag, New York-Berlin-Heidelberg, 1999, 34–65.
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This research was supported by the National Natural Science Foundation of China under Grant No. 12161057.
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Xiao, S., Zeng, G. The Invertibility of Rational Univariate Representations. J Syst Sci Complex 35, 2430–2451 (2022). https://doi.org/10.1007/s11424-022-1070-3
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DOI: https://doi.org/10.1007/s11424-022-1070-3