Characteristic Set Method

Characteristic Set Method: Algebraic Case

• S.C. Chou and X.S. Gao, Ritt-Wu's Decomposition Algorithm and Geometry Theorem Proving in Proceedings of CADE--10, Lecture Notes in Artificial Intelligence 449, Springer--Verlag, 1990, 207--220.
• X.S. Gao and S. C. Chou, The Dimension of Ascending Chains, Chinese Science Bulletin, Vol. 38, No.5, 1993, 396--399.
• X.S. Gao and S. C. Chou, On the Theory of Resolvents and Its Applications, Sys. Sci. and Math. Sci., 12, Suppl., 17--30, 1999,
• X.S. Gao and K. Jiang, Order in Solving Polynomial Equations, Computer Mathematics, Proc. of ASCM'2000, pp. 308-318, World Scientific, 2000.
• SC Chou and XS Gao, An Algebraic System Based on the Characteristic Set Method, in Proceedings of International Workshop on Mathematics Mechanization, Beijing, International Academic Publishers, p.1--17, 1992.

Characteristic Set Method: Parametric Equation Systems

• X.S. Gao and S.C. Chou, Solving Parametric Algebraic Systems, in the Proceedings of ISSAC'92 , Berkeley, 335--341, ACM Press, 1992.
• X.S. Gao and S. C. Chou, A Zero Structure Theorem for Differential Parametric Systems, Journal of Symbolic Computation, Vol. 16, 585-595, 1993.
• X.S. Gao and D.K. Wang, Zero Decomposition Theorems for Counting the Number of Solutions for Parametric Equation Systems, in Computer Mathematics, Proc. of ASCM2003, (eds. Z. Li and W. Sit), 130-145, World Scientific, Singapore, 2003.

Characteristic Set Method: Boolean polynomials and finite fields

• F. Chai, X.S. Gao, and C. Yuan, A Characteristic Set Method for Solving Boolean Equations and Applications in Cryptanalysis of Stream Ciphers, Journal of Systems Science and Complexity, 21(2), 191-208, 2008.
• Xiao-Shan Gao and Zhenyu Huang, A Characteristic Set Method for Equation Solving in Finite Fields, MM-Preprints, Vol. 26, 77-92, January, 2008. Accepted by JSC.
• S.W. Zhao and X.S. Gao, Minimal Achievable Approximation Ratio for MAX-MQ in Finite Fields, Theoretical Computer Science, 410(21-23), 2285-2290, 2009.

Real Roots Isolation of Triangular Systems

• J.S. Cheng, X.S. Gao, and C.K. Yap, Complete Numerical Isolation of Real Roots in Zero-dimensional Triangular Systems, Journal of Symbolic Computation£¬ 44(7)£¬ 768-785, 2009.
• J.S. Cheng, X.S. Gao, J. Li, Root Isolation for Bivariate Polynomial Systems with Local Generic Position , Proc. ISSAC 2009, 103-109, ACM Press, New York, 2009.

Characteristic Set Method: Differential Case

• S. C. Chou and X.S. Gao, Automated Reasoning in Differential Geometry and Mechanics: Part I. An Improved Version of Ritt--Wu's Decomposition Algorithm, Journal of Automated Reasoning, 10: 161--172, 1993.
• X.S. Gao, Implicitization for Differential Rational Parametric Equations, J. of Symbolic Computation, 811-824, 36(5), 2003..
• Y. Chen and X.S. Gao, Involutive Bases of Algebraic Partial Differential Equation Systems, Science in China (A), 33(2), 97-113, 2003.
• T. Chaolu and X.S. Gao, Nearly Differential Characteristic Set for Differential Polynomial System, Acta Mathematica Sinica, 45(6), 1041-1050, 2002.

Characteristic Set Method: Difference Case

• X.S. Gao, Y. Luo, and C. Yuan, A Characteristic Set Method for Difference Polynomial Systems, Journal of Symbolic Computation, 44(3), 242-260, 2009.
• X.S. Gao, J. Van der Hoeven, C.M. Yuan, G.L. Zhang, Characteristic set method for differential¨Cdifference polynomial systems, Journal of Symbolic Computation, 44(9), 2009, 1137-1163.
• X.S. Gao, C. Yuan, and G. Zhang, Ritt-Wu's Characteristic Set Method for Ordinary Difference Polynomial Systems with Arbitrary Ordering, Acta Mathematica Scientia, 29(3,4), 1063-1080, 2009.
• X.S. Gao and C. Yuan, Resolvent Systems of Difference Polynomial Ideals, Proc. ISSAC 2006, ACM Press, New York, 2006.