Wei Pan
ABSTRACT
This paper presents a method for computing uniform Gröbner bases for certain ideals generated by polynomials with parametric exponents. The method proceeds by replacing monomials involving parametric exponents in the generators of an ideal with new variables, computing the reduced Gröbner basis for the resulting ideal with respect to a special monomial order, and then verifying whether the leading monomial ideal of the Gröbner basis satisfies some consistency conditions according to two criteria (of which one is derived from Buchberger graphs). When the consistency conditions are verified, a uniform Gröbner basis for the original ideal is obtained by substituting the new variables back to original monomials. The effectiveness and practical value of the method are demonstrated by its application to a family of ideals coming from the modeling of biological systems.
- Bayer, D., Galligo, A., Stillman, M.: Gröbner bases and extension of scalars. In: Computational Algebraic Geometry and Commutative Algebra (Eisenbud, D., Robbiano, L., eds.), pp. 198--215. Cambridge University Press, Cambridge, 1993.]]Google Scholar
- Buchberger, B.: A criterion for detecting unnecessary reductions in the construction of Gröbner bases. In: Proc. EUROSAM '79 (Marseille, France, June 1979), LNCS 72, pp. 3--21. Springer, Berlin Heidelberg, 1979.]] Google ScholarDigital Library
- Cinquin, O., Demongeot, J.: Positive and negative feedback: Striking a balance between necessary antagonists. J. Theor. Biol. 216: 229--241, 2002.]]Google ScholarCross Ref
- Cox, D., Little, J., O'Shea, D.: Ideals, Varieties and Algorithms. Springer, New York, 1992.]] Google ScholarDigital Library
- Cox, D., Little, J., O'Shea, D.: Using Algebraic Geometry. GTM 185. Springer, New York, 1998.]]Google Scholar
- Diestel, R.: Graph Theory. Springer, New York, 1997.]]Google Scholar
- Eisenbud, D.: Commutative Algebra. GTM 150. Springer, New York, 1994.]]Google Scholar
- Hong, H.: Gröbner basis under composition II. In: Proc. ISSAC '96 (Zurich, Switzerland, July 24-26, 1996), pp. 79--85. ACM Press, New York, 1996.]] Google ScholarDigital Library
- Kalkbrener, M.: On the stability of Gröbner bases under specializations. J. Symb. Comput. 24: 51--58, 1997.]] Google ScholarDigital Library
- Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. GTM 227. Springer, New York, 2004.]]Google Scholar
- Mishra, B.: Algorithmic Algebra. Springer, New York, 1993.]] Google ScholarDigital Library
- Pan, W.: Uniform free resolutions of monomial ideals. Preprint (submitted for publication), University of Science and Technology of China, China, 2006.]]Google Scholar
- Wang, D.: The projection property of regular systems and its application to solving parametric polynomial systems. In: Algorithmic Algebra and Logic (Passau, Germany, April 3-6, 2005), pp. 269--274. Herstellung und Verlag, Norderstedt, 2005.]]Google Scholar
- Wang, D., Xia, B.: Algebraic analysis of stability for some biological systems. In: Algebraic Biology 2005 - Computer Algebra in Biology (Tokyo, Japan, November 28-30, 2005), pp. 75--83. Universal Academy Press, Inc., Tokyo, 2005.]]Google Scholar
- Weispfenning, V.: Gröbner bases for binomials with parametric exponents. Preprint, Universität Passau, Germany, 2004.]]Google Scholar
- Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comput. 14: 1--29, 1992.]] Google ScholarDigital Library
- Yokoyama, K.: On systems of algebraic equations with parametric exponents. In: Proc. ISSAC 2004 (Santander, Spain, July 4--7, 2004), pp. 312--317. ACM Press, New York, 2004.]] Google ScholarDigital Library
- Yokoyama, K.: On systems of algebraic equations with parametric exponents II. Presented at ACA 2005 (Nara, Japan, July 31 - August 3, 2005) and submitted for publication.]]Google Scholar
Index Terms
- Uniform Gröbner bases for ideals generated by polynomials with parametric exponents
Recommendations
Fast computation of Gröbner bases of ideals of F[x, y]
ISIT'09: Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4This paper provides a fast algorithm for Gröbner bases of ideals of F[x, y] over a field F. We show that only the S-polynomials of neighbor pairs of a strictly ordered finite generating set are needed in the computing of a Gröbner bases of the ideal. It ...
Janet-like gröbner bases
CASC'05: Proceedings of the 8th international conference on Computer Algebra in Scientific ComputingWe define a new type of Gröbner bases called Janet-like, since their properties are similar to those for Janet bases. In particular, Janet-like bases also admit an explicit formula for the Hilbert function of polynomial ideals. Cardinality of a Janet-...
Equivariant Gröbner Bases of Symmetric Toric Ideals
ISSAC '16: Proceedings of the ACM on International Symposium on Symbolic and Algebraic ComputationIt has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have finite Gröbner ...
Comments