Abstract
This paper provides a new, geometric perspective to study successive difference substitutions, and proves that the sequence of the successive difference substitution sets is not convergent. An interesting result that a given k-dimensional rational hyperplane can be transformed to a k-dimensional coordinate hyperplane of new variables by finite steps of successive difference substitutions is presented. Moreover, a sufficient condition for the sequence of the successive difference substitution sets of a form being not terminating is obtained. That is, a class of polynomials which cannot be proved to be positive semi-definite by the successive difference substitution method are obtained.
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References
Lasserre J B. Global optimization with polynomials and the problem of moments. SIAM J Opt, 2001, 11: 796–817
Parrilo P A. Semidefinite programming relaxations for semialgebraic problems. Math Prog Ser B, 2003, 96: 293–320
Henrion D, Garulli A, eds. Positive Polynomials in Control. Vol. 312. New York: Springer, 2005
Collins G E, Hong H. Partial cylindrical algebraic decomposition for quantifier elimination. J Symb Comput, 1991, 12: 299–328
Yang L, Hou X R, Xia B C. A complete algorithm of automated discovering for a class of inequality type theorems. Sci China Ser F-Inf Sci, 2001, 44: 33–49
Pólya G, Szego G. Problems and Theorems in Analysis. Vol. 2. New York: Springer-Verlag, 1972
Hardy G H, Littlcwood J E, Pólya G. Inequalities. Cambridge: Cambridge University Press, 1952
Catlin D W, D’Angelo J P. Positivity conditions for bihomogeneous polynomials. Math Res Lett, 1997, 4: 555–567
Handelman D. Deciding eventual positivity of polynomials. Ergod Th & Dynam, 1986, 6: 57–59
Yang L. Solving harder problems with lesser mathematic. In: Proceedings of the 10th Asian Technology Conference in Mathematics. Singapore: National Institute of Education, 2005. 37–46
Yang L, Xia B C. Automated Proving and Discovering on Inequalities (in Chinese). Beijing: Science Press, 2008. 163–175
Yao Y. Termination of the sequence of SDS sets and machine decision for positive semi-definite forms. arXiv: 0904.4030
Chen L Y, Zeng Z B. Which symmetric homogeneous polynomials can be proved positive semi-definite by difference substitution method? In: ASCM 2007. Singapore: National Institute of Education, 2007. 57–71
Zhang J C, Liang Y Z, Zhen B. Hybrid high performance model of computer algebra system (in Chinese). J Comput Appl, 2007, 27: 2834–2837
Spanier E H. Algebraic Topology. New York: Springer-Verlag, 1966
Munkres J R. Elements of Algebraic Topology. Reading, MA: Addison Wesley Publishing Company, 1984
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Hou, X., Xu, S. & Shao, J. Some geometric properties of successive difference substitutions. Sci. China Inf. Sci. 54, 778–786 (2011). https://doi.org/10.1007/s11432-010-4178-3
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DOI: https://doi.org/10.1007/s11432-010-4178-3