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Algorithmic criteria for permutation orthogonal systems over finite fields

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Chinese Science Bulletin 1998年第16期43卷 1326-1331页

作者： LU Peizhong Institute of Systems Science Chinese Academy of SciencesBeijing 100080China

Two criteria for orthogonal systems over finite fields are presented. The first one is an algorithm by use of Gr*ibner bases. The second one is a formula of the numbers of adjoint morphisms.

关键词： permutation polynomld orthogonal system algebraic variety Grobner basis

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

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Science China Mathematics 2012年第12期55卷 2433-2446页

作者： YANG LiLi YANG Lu YU ZhiHeng ZHANG WeiNian Department of Mathematics Sichuan University Chengdu 610064 China Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 200062 China

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.

关键词： iterative root polynomial algebraic variety Sylvester resultant elimination

Hilbert's Sixteenth Problem and Its Generalization

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Journal of Shanghai University(English Edition) 2001年第2期5卷 96-103页

作者： SHENG Ping xing Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200436, China Department of Mathematics College of Sciences Shanghai University Shanghai China

This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree n are discussed. The maximum number of sheets for an algebraic equation of degree n and the maximum number of limit cycles for a planar algebraic autonomous system are achieved. The author also considers different generalizations and some related problems.

关键词： algebraic curve sheet limit cycle algebraic variety irreducible and prime polynomial cardinal number configuration