The Invertibility of Rational Univariate Representations

doi:10.1007/s11424-022-1070-3

Journal of Systems Science and Complexity ›› 2022, Vol. 35 ›› Issue (6): 2430-2451.DOI: 10.1007/s11424-022-1070-3

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The Invertibility of Rational Univariate Representations

XIAO Shuijing, ZENG Guangxing

Department of Mathematics, Nanchang University, Nanchang 330031, China

Received:2021-03-13 Revised:2021-07-02 Online:2022-11-25 Published:2022-12-23
Supported by:
This research was supported by the National Natural Science Foundation of China under Grant No. 12161057.

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XIAO Shuijing, ZENG Guangxing. The Invertibility of Rational Univariate Representations[J]. Journal of Systems Science and Complexity, 2022, 35(6): 2430-2451.

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.

Key words: Invertibility, polynomial ideal, rational univariate representation (RUR), Wu's method, zero-dimensional

[1] Rouillier F, Solving zero-dimensional systems through the rational univariate representation, AAECC, 1995, 9: 433–461.
[2] 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).
[3] 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).
[4] Xiao S J and Zeng G X, Solving the equality-constrained minimization problem of polynomial functions, J. Global Optimization, 2019, 75: 683–733.
[5] 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.
[6] 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.
[7] Nagata M, Field Theory, Marcel Dekker, Inc., New York, 1977.
[8] Becker T, Weispfenning V, and Kredel H, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer-Verlag, New York-Berlin-Heidelberg, 1993.
[9] Atiyah M F and MacDonald J G, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.
[10] Jacobson N, Basic Algebra I, 2nd Edition, W. H. Freeman and Company, New York, 1985.
[11] Mishra B, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer-Verlag, New York-Berlin-Heidelberg, 1993.
[12] 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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