Üzerinde Sıfırlanan Polinomların İdealinin Açık Gröbner Tabanı

Year 2017, Volume: 7 Issue: 2, 349 - 351, 01.06.2017

Abdullah Çağman

Abstract

Sıfırlanan polinomlar, katsayı halkası üzerinde tanımlanan polinom halkalarının bir idealini oluştururlar. Bu makalede m,l ≠1 olmak üzere Zidealinin açık minimal güçlü Gröbner tabanını vereceğiz. İspatımız tamamen kobinasyonel yönteme dayalı olacaktır

Keywords

Gröbner tabanı, Polinom halkası, Sıfırlanan ideal, Sıfırlanan polinom

Explicit Gröbner Basis of the Ideal of Vanishing Polynomials over Z2×Z2

Abstract

Vanishing polynomials form an ideal of polynomial ring over the coefficient ring. In this paper, we give some vanishing polynomials of the polynomial ring Zm×Zl[x1,x2,…,xn] where m,l ≠1 and an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials of the ring Z2×Z2[x]. Our proof is based fully on a combinatorial way.

Keywords

Vanishing polynomial, Polynomial ring, Vanishing Ideal, Gröbner Basis

References

• Aoki, S., Hibi, T., Ohsugi, H., Takemura, A. 2010. Markov basis and Gröbner basis of Segre– Veronese configuration for testing independence in group-wise selections. Ann. Inst. Stat. Math., 62: 299– 321.
• Buchberger, B. 1965. An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal, PhD thesis, Univ. of Innsbruck (Austria), 36 pp.
• Conti, P., Traverso, C. 1991. Buchberger algorithm and integer progamming, In: Mattson, H., Mora, T., Rao, T. [eds.], Applied Algebra Algebraic Algorithms and Error Correcting Codes. Lecture Notes in Computer Science, Springer, Berlin, vol. 539, pp. 130-139.
• De Loera, Jesşs A. 1995. Gröbner bases and graph colorings. Beiträge Algebra Geom., 1995: 36 (1): 89-96.
• Grayson, DR., Stillman, ME. 01 August 2016. Macaulay 2, a software system for research in algebraic geometry. http:// www.math.uiuc.edu/Macaulay2/
• Greuel, GM., Seelisch, F., Wienand, O. 2011. The Gröbner basis of the ideal of vanishing polynomials. J. Symbolic Comput., 46: 561-570.
• Greuel, GM., Wedler, M., Wienand, O., Brickenstein, M., Dreyer, A. 2008. New developments in the theory of Groebner bases and applications to formal verification. J. Pure Appl. Algebra, 213(8): 1612-1635.
• Hironaka, H. 1964. Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math., 79: 109– 203.
• Macaulay, F.S. 1927. Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc., 26: 531- 555.
• Shekhar, N., Kalla, P., Enescu, F., Gopalakrishnan, S. 2005. Equivalence verification of polynomial datapaths with fixed- size bitvectors using finite ring algebra. In the proceedings of the 2005 IEEE/ACM International Conference on Computeraided Design, pp: 291– 296.
• Sturmfels, B. 1996. Gröbner Bases and Convex Polytopes, Amer. Math. Soc., Providence, 162 pp.
• Wienand, O., Wedler, M., Stoffel, D., Kunz, W., Greuel, GM. 2008. An algebraic approach for proving data correctness in arithmetic data paths. In the proceedings of the 20th International Conference on Computer Aided Verification, pp: 473– 486.