平面三次多项式的分类

Abstract

Abstract:

In this paper, we consider cubic polynomial normal forms on $\mathbb{R}^2$. By employing the theory of polynomial algebra to find the minimal irreducible decomposition of the corresponding varieties, we obtain smooth classifications of a kind of planar cubic polynomials via global conjugations. Our results are also applied to study the problems of iterative roots and embedding flows on $\mathbb{R}^2$.

Key words: planar cubic polynomial, conjugacy, smooth normal form, minimal irreducible decomposition, iterative root

CLC Number:

O192

Li Lin,Liu Lingling,Yu Zhiheng. On Classification of Planar Cubic Polynomials[J].Acta mathematica scientia,Series A, 2025, 45(2): 554-566.

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References 41

[1]	Arango J, Gómez A. Diffeomorphisms as time one maps. Aequationes Math, 2002, 64: 304-314
[2]	Arcet B, Giné J, Romanovski V. Linearizability of planar polynomial Hamiltonian systems. Nonlinear Anal-Real, 2022, 63: Article 103422
[3]	Arnold V I, Yu S I. Ordinary Differential Equations. Berlin: Springer-Verlag, 1988
[4]	Ashkenazi M, Chow S. Normal forms near critical points for differential equations and maps. IEEE Trans Circuits Systems, 1988, 35: 850-862
[5]	Becker T, Weispfenning V. Gröbner Bases:A Computational Approach to Commutative Algebra. New York: Springer, 1993
[6]	Belitskii G R. Equivalence and normal forms of germs of smooth mappings. Russ Math Surv, 1978, 33: 107-177
[7]	Bronstein I U, Kopanskii A Y. Smooth Invariant Manifolds and Normal Forms. Singapore: World Scientific, 1994
[8]	Buchberger B. Ein Algorithmus Zum Auffinden Der Basiselemente Des Restklassenringes Nach Einem Nulldimensional Polynomideal. Austria: Universität Innsbruck, 1965
[9]	Chen G. Further reduction of normal forms for vector fields. Numerical Algorithms, 2001, 27: 1-33
[10]	Chen G, Dora J. Further reductions of normal forms for dynamical systems. J Differ Equ, 2000, 166: 79-106
[11]	Chen G, Dora J. Normal forms for differentiable maps near a fixed point. Numerical Algorithms, 1999, 22: 213-230
[12]	Chen G, Dora J. An algorithm for computing a new normal form for dynamical systems. J Symbolic Computation, 2000, 29: 393-418
[13]	Chen X, Shi Y, Zhang W. Planar quadratic degree-preserving maps and their iteration. Result Math, 2009, 55: 39-63
[14]	Chen G, Wang D, Yang J. Unique normal forms for Hopf-zero vector fields. C R Acad Sci Paris, 2003, 336: 345-348
[15]	Cuong L, Doan T, Siegmund S. A sternberg theorem for nonautonomous differential equations. J Dyn Diff Equ, 2019, 31: 1279-1299
[16]	Gelfand M, Kapranov M M, Zelevinsky A V. Discriminants, Resultants and Multidimensional Determinant. Boston: Birkhäuser, 1994
[17]	Gramchev T, Walcher S. Normal forms of maps: formal and algebraic aspects. Acta Appl Math, 2005, 87: 123-146
[18]	Guo L, Ren Z. Polynomial normal forms for some germs of nonstrongly 1-resonant diffeomorphisms. Int J Bifurcation and Chaos, 2015, 25: Article 1550174
[19]	Han M. Conditions for a diffeomorphism to be embedded in a $C^r$ flow. Acta Math Sinica, 1988, 4: 111-123
[20]	Han Z, Zhang Z, Zheng Y. New normal for nonlinear system (II)-controllable form. Acta Math Sci, 1993, 13B(2): 139-146
[21]	Ichikawa F. Finitely determined singularities of formal vector fields. Invent Math, 1982, 66: 199-214
[22]	Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. New York: Cambridge University Press, 1995
[23]	Kuczma M. Functional Equations in a Single Variable. Warszawa: Polish Scientific Publishers, 1968
[24]	Kuczma M, Choczewski B, Ger R. Iterative Functional Equations. Cambridge: Cambridge University Press, 1990
[25]	Lam P F. Embedding a differentiable homeomorphism in a flow subject to a regularity condition on the derivatives of the positive transition homeomorphisms. J Differ Equ, 1978, 30: 31-40
[26]	Li J, Kou L, Wang D, Zhang W. Unique normal form and the associated coefficients for a class of three-dimensional nilpotent vector fields. Int J Bifurcation and Chaos, 2017, 27(14): 1750224
[27]	李伟固. 正规形理论及其应用. 北京: 科学出版社, 2000
	Li W G. Normal Form Theory and Its Application. Beijing: Science Press, 2000
[28]	Li W, Llibre J, Wu H. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2015, 36(1): 345-360
[29]	Liu L, Tang Y, Zhang W. Versal unfolding of homogeneous cubic degenerate centers in strong monodromic family. J Differ Equ, 2021, 283: 136-162
[30]	Rayskin V. Hölder linearization. J Differ Equ, 1998, 147: 271-284
[31]	Ren Z, Xia L, Yang J. On classification of the Poincaré type maps on $\mathbb{R}^3$. Bull Sci Math, 2015, 139(5): 582-598
[32]	Ren Z, Yang J, Yu F. On conjugating equivalence of 0-resonant diffeomorphisms on $\mathbb{R}^3$. Int J Bifurcation and Chaos, 2013, 23(6): Article 1350100
[33]	Ren Z, Yan S, Yang J. Finite determinacy and polynomial normal forms for diffeomorphisms near a strongly 1-resonant fixed point. J Math Anal Appl, 2014, 413(1): 476-481
[34]	Rodrigues H M, Sola-Morales J. Linearization of class $C^1$ for contractions on Banach spaces. J Differ Equ, 2004, 201: 351-382
[35]	Romanovski V, Shafer D. The Center and Cyclicity Problems:A Computational Algebra Approach. Boston: Birkhäuser, 2009
[36]	Strien S V. Smooth linearization of hyperbolic fixed points without resonance conditions. J Differ Equ, 1990, 85(1): 66-90
[37]	Takens F. Singularities of vector fields. Publ Math de lÍnstit des Hautes Etudes Sci, 1974, 43: 47-100
[38]	Takens F. Normal forms for certain singularities of vector fields. Ann Inst Fourier, 1973, 23(2): 163-195
[39]	Turqui A, Dali D. Normal forms of planar polynomial differential systems. Qual Theory Dyn Syst, 2019, 18: 11-33
[40]	Yang J. Polynomial normal forms for vector fields on $\mathbb{R}^3$. Duke Math J, 2001, 106(1): 1-18
[41]	Yu Z, Yang L, Zhang W. Discussion on polynomials having polynomial iterative roots. J Symbolic Comput, 2012, 47(10): 1154-1162

On Classification of Planar Cubic Polynomials

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