Abstract
Pan and Wang presented a method for computing uniform Gröbner bases for certain ideals generated by polynomials with parametric exponents in 2006, and two criteria were proposed to determine if a uniform Gröbner basis can be obtained. This paper gives a new algorithmic approach for computing the uniform Gröbner basis such that Pan and Wang’s method could be concluded as a special case. The authors use the method of reduced term order under ring homomorphism to get the reduced uniform Gröbner basis. Also the authors point and correct a mistake in Pan and Wang’s method. The result is a generalization of approach of Pan and Wang and one could compute the uniform Gröbner basis more efficiently by the new approach.
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Pan W and Wang D M, Uniform Gröbner bases for ideals generated by polynomials with parametric exponents, Proceedings ISSAC 2006, ACM Press, 2006, 269–276.
Cinquin O and Demongeot J, Positive and negative feedback: Striking a balance between necessary antagonists, J. Theor. Biol., 2002, 216: 229–241.
Weispfenning V, Gröbner bases for binomials with parametric exponents, Proceedings of CASA 2004, Springer-Verlag, 2004, 467–478.
Yokoyama K, On systems of algebraic equations with parametric exponents, Proceedings ISSAC 2004, ACM Press, 2004, 312–317.
Yokoyama K, On systems of algebraic equations with parametric exponents II, Presented at ACA 2005 (Nara, Japan, July 31–August 3), 2005.
Zhou M and Winkler F, Gröbner bases in difference-differential modules and difference-differential dimension polynomials, Science China Math., 2008, 51(9): 1731–1752.
Zhou M and Winkler F, Computing difference-differential dimension polynomials by relative Gröbner bases in difference-differential modules, J. Symb. Comput., 2008, 43: 726–745.
Weispfenning V, Comprehensive Gröbner bases, J. Symb. Comput., 1992, 14: 1–29.
Wang D M and Xia B C, Algebraic analysis of stability for some biological systems, Algebraic Biology 2005 - Computer Algebra in Biology (Tokyo, Japan, November 28–30), Universal Academy Press, 2005, 75–83.
Wang D M, The projection property of regular systems and its application to solving parametric polynomial systems, Algorithmic Algebra and Logic (Passau, Germany, April 3–6), Herstellung und Verlag}, Norderstedt, 2005, 269–274.
Ma X D, Sun Y, and Wang D K, On computing Gröbner bases in rings of differential operators, Science China Math., 2011, 54(6): 1077–1087.
Zhou M and Winkler F, Gröbner bases in difference-differential modules, Proceedings ISSAC 2006, ACM Press, 2006, 353–360.
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This research was supported by the National Natural Science Foundation of China under Grant No. 11271040 and Science and Technology Foundation of GuiZhou Province LKM[2013]16.
This paper was recommended for publication by Editor ZHANG Yang.
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Liu, L., Zhou, M. On computing uniform Gröbner bases for ideals generated by polynimials with parametric exponents. J Syst Sci Complex 29, 850–864 (2016). https://doi.org/10.1007/s11424-016-4297-z
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DOI: https://doi.org/10.1007/s11424-016-4297-z