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Topological Classifications for a Class of 2-dimensional Quadratic Mappings and an Application to Iterative Roots

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Abstract

Usually, it is difficult to find a topological relation among high dimensional mappings because of the complexity of existence conditions for conjugacies. In this paper, we investigate the conjugation for a class of 2-dimensional quadratic mappings via the theory of polynomial algebra. Two kinds of conjugate classifications are given. Finally, an application to the problem of iterative roots is also presented.

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Acknowledgements

The authors are very grateful to the editor and reviewers for their carefully checking and helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, 611756, Chengdu, Sichuan, People’s Republic of China

    Zhiheng Yu

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, 314001, Jiaxing, Zhejiang, People’s Republic of China

    Lin Li

  3. School of Sciences, Southwest Petroleum University, 610500, Chengdu, Sichuan, People’s Republic of China

    Lingling Liu

  4. Institute for Artificial Intelligence, Southwest Petroleum University, 610500, Chengdu, Sichuan, People’s Republic of China

    Lingling Liu

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  1. Zhiheng Yu
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  2. Lin Li
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Correspondence to Lin Li.

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This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grant #LY18A010017, the National Science Foundation of China Grant #11671061, 11501475, 11771308, 11871041 and the Fundamental Research Funds for the Central Universities under grant 2682018CX63.

Appendix: Singular Worksheet for Case (i) of Lemma 1

Appendix: Singular Worksheet for Case (i) of Lemma 1

> LIB “primdec.lib”;

> ring r= (0),(c3,c4,c5,d3,d4,d5,b5,b4,b3,b2,b1,a5,a4,

a3,a2,a1,u),(dp);

> r;

> option(prot);

> short=0;

> // define polynomials in system APS

> poly f1 = a3*c3⌃2+a4*c3*d3+a5*d3⌃2;

> poly f2 = 2*a3*c3*c4+a4*c3*d4+a4*c4*d3+2*a5*d3*d4;

> poly f3 = 2*a3*c3*c5+a3*c4⌃2+a4*c3*d5+a4*c4*d4+a4*c5*d3

+2*a5*d3*d5 +a5*d4⌃2;

> poly f4 = 2*a3*c4*c5+a4*c4*d5+a4*c5*d4+2*a5*d4*d5;

> poly f5 = a3*c5⌃2+a4*c5*d5+a5*d5⌃2;

> poly f6 = 2*a3*b2*c3-a4*b1*c3+a4*b2*d3-2*a5*b1*d3;

> poly f7 = a1*a4*c3+2*a1*a5*d3-2*a2*a3*c3-a2*a4*d3

+2*a3*b2*c4-a4*b1*c4 +a4*b2*d4-2*a5*b1*d4;

> poly f8 = a1*a4*c4+2*a1*a5*d4-2*a2*a3*c4-a2*a4*d4

+2*a3*b2*c5-a4*b1*c5 +a4*b2*d5-2*a5*b1*d5;

> poly f9 = a1*a4*c5+2*a1*a5*d5-2*a2*a3*c5-a2*a4*d5;

> poly f10 = a1⌃3*b2⌃2*c3-2*a1⌃2*a2*b1*b2*c3

+a1⌃2*a2*b2⌃2*d3 +a1*a2⌃2*b1⌃2*c3

-2*a1*a2⌃2*b1*b2*d3+a2⌃3*b1⌃2*d3+a3*b2⌃2

-a4*b1*b2+a5*b1⌃2;

> poly f11 = a1⌃3*b2⌃2*c4-2*a1⌃2*a2*b1*b2*c4

+a1⌃2*a2*b2⌃2*d4 +a1*a2⌃2*b1⌃2*c4

-2*a1*a2⌃2*b1*b2*d4+a2⌃3*b1⌃2*d4+a1*a4*b2

-2*a1*a5*b1-2*a2*a3*b2+a2*a4*b1;

> poly f12 = a1⌃3*b2⌃2*c5-2*a1⌃2*a2*b1*b2*c5

+a1⌃2*a2*b2⌃2*d5+a1*a2⌃2*b1⌃2*c5-2*a1*a2⌃2*b1*b2*d5

+a2⌃3*b1⌃2*d5+a1⌃2*a5

-a1*a2*a4+a2⌃2*a3;

> poly f13 = b3*c3⌃2+b4*c3*d3+b5*d3⌃2;

> poly f14 = 2*b3*c3*c4+b4*c3*d4+b4*c4*d3+2*b5*d3*d4;

> poly f15 = 2*b3*c3*c5+b3*c4⌃2+b4*c3*d5+b4*c4*d4

+b4*c5*d3+2*b5*d3*d5 +b5*d4⌃2;

> poly f16 = 2*b3*c4*c5+b4*c4*d5+b4*c5*d4+2*b5*d4*d5;

> poly f17 = b3*c5⌃2+b4*c5*d5+b5*d5⌃2;

> poly f18 = b1*b4*c3+2*b1*b5*d3-2*b2*b3*c3-b2*b4*d3;

> poly f19 = a1*b4*c3+2*a1*b5*d3-2*a2*b3*c3-a2*b4*d3

-b1*b4*c4-2*b1*b5*d4 +2*b2*b3*c4+b2*b4*d4;

> poly f20 = a1*b4*c4+2*a1*b5*d4-2*a2*b3*c4-a2*b4*d4

-b1*b4*c5-2*b1*b5*d5 +2*b2*b3*c5+b2*b4*d5;

> poly f21 = a1*b4*c5+2*a1*b5*d5-2*a2*b3*c5-a2*b4*d5;

> poly f22 = a1⌃2*b1*b2⌃2*c3+a1⌃2*b2⌃3*d3

-2*a1*a2*b1⌃2*b2*c3-2*a1*a2*b1*b2⌃2*d3+a2⌃2*b1⌃3*c3

+a2⌃2*b1⌃2*b2*d3+b1⌃2*b5

-b1*b2*b4+b2⌃2*b3;

> poly f23 = a1⌃2*b1*b2⌃2*c4+a1⌃2*b2⌃3*d4

-2*a1*a2*b1⌃2*b2*c4-2*a1*a2*b1*b2⌃2*d4+a2⌃2*b1⌃3*c4

+a2⌃2*b1⌃2*b2*d4-2*a1*b1*b5

+a1*b2*b4+a2*b1*b4-2*a2*b2*b3;

> poly f24 = a1⌃2*b1*b2⌃2*c5+a1⌃2*b2⌃3*d5

-2*a1*a2*b1⌃2*b2*c5-2*a1*a2*b1*b2⌃2*d5+a2⌃2*b1⌃3*c5

+a2⌃2*b1⌃2*b2*d5+a1⌃2*b5

-a1*a2*b4+a2⌃2*b3;

> poly f25 = a3⌃2*c3+a3*b3*c4+b3⌃2*c5;

> poly f26 = 2*a3*a4*c3+a3*b4*c4+a4*b3*c4+2*b3*b4*c5;

> poly f27 = 2*a3*a5*c3+a3*b5*c4+a4⌃2*c3+a4*b4*c4

+a5*b3*c4+2*b3*b5*c5 +b4⌃2*c5;

> poly f28 = 2*a4*a5*c3+a4*b5*c4+a5*b4*c4+2*b4*b5*c5;

> poly f29 = a5⌃2*c3+a5*b5*c4+b5⌃2*c5;

> poly f30 = 2*a1*a3*c3+a1*b3*c4+a3*b1*c4+2*b1*b3*c5;

> poly f31 = 2*a1*a4*c3+a1*b4*c4+2*a2*a3*c3+a2*b3*c4

+a3*b2*c4+a4*b1*c4+2*b1*b4*c5+2*b2*b3*c5;

> poly f32 = 2*a1*a5*c3+a1*b5*c4+2*a2*a4*c3+a2*b4*c4

+a4*b2*c4+a5*b1*c4+2*b1*b5*c5+2*b2*b4*c5;

> poly f33 = 2*a2*a5*c3+a2*b5*c4+a5*b2*c4+2*b2*b5*c5;

> poly f34 = a1⌃3*b2*c3-a1⌃2*a2*b1*c3+a1⌃2*b1*b2*c4

-a1*a2*b1⌃2*c4+a1*b1⌃2*b2*c5-a2*b1⌃3*c5-a2*b3+a3*b2;

> poly f35=2*a1⌃2*a2*b2*c3+a1⌃2*b2⌃2*c4

-2*a1*a2⌃2*b1*c3+2*a1*b1*b2⌃2*c5

-a2⌃2*b1⌃2*c4-2*a2*b1⌃2*b2*c5-a2*b4+a4*b2;

> poly f36 = a1*a2⌃2*b2*c3+a1*a2*b2⌃2*c4+a1*b2⌃3*c5

-a2⌃3*b1*c3-a2⌃2*b1*b2*c4-a2*b1*b2⌃2*c5-a2*b5+a5*b2;

> poly f37 = a3⌃2*d3+a3*b3*d4+b3⌃2*d5;

> poly f38 = 2*a3*a4*d3+a3*b4*d4+a4*b3*d4+2*b3*b4*d5;

> poly f39 = 2*a3*a5*d3+a3*b5*d4+a4⌃2*d3+a4*b4*d4

+a5*b3*d4+2*b3*b5*d5 +b4⌃2*d5;

> poly f40 = 2*a4*a5*d3+a4*b5*d4+a5*b4*d4+2*b4*b5*d5;

> poly f41 = a5⌃2*d3+a5*b5*d4+b5⌃2*d5;

> poly f42 = 2*a1*a3*d3+a1*b3*d4+a3*b1*d4+2*b1*b3*d5;

> poly f43 = 2*a1*a4*d3+a1*b4*d4+2*a2*a3*d3+a2*b3*d4

+a3*b2*d4+a4*b1*d4+2*b1*b4*d5+2*b2*b3*d5;

> poly f44 = 2*a1*a5*d3+a1*b5*d4+2*a2*a4*d3+a2*b4*d4

+a4*b2*d4+a5*b1*d4+2*b1*b5*d5+2*b2*b4*d5;

> poly f45 = 2*a2*a5*d3+a2*b5*d4+a5*b2*d4+2*b2*b5*d5;

> poly f46 = a1⌃3*b2*d3-a1⌃2*a2*b1*d3+a1⌃2*b1*b2*d4

-a1*a2*b1⌃2*d4 +a1*b1⌃2*b2*d5-a2*b1⌃3*d5+a1*b3-a3*b1;

> poly f47 = 2*a1⌃2*a2*b2*d3+a1⌃2*b2⌃2*d4

-2*a1*a2⌃2*b1*d3+2*a1*b1*b2⌃2*d5

-a2⌃2*b1⌃2*d4-2*a2*b1⌃2*b2*d5+a1*b4-a4*b1;

> poly f48 = a1*a2⌃2*b2*d3+a1*a2*b2⌃2*d4+a1*b2⌃3*d5

-a2⌃3*b1*d3-a2⌃2*b1*b2*d4-a2*b1*b2⌃2*d5+a1*b5-a5*b1;

> ideal APS=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,

f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,

f28,f29,f30,f31,f32,f33,f34,f35,f36,f37,f38,f39,

f40,f41,f42,f43,f44,f45,f46,f47,f48;

> // compute Gröbner basis of APS

> ideal G=slimgb(APS);

> // find the minimal irreducible decomposition for Gröbner basis G

> minAssGTZ(G);

> // define polynomials in system \({\widetilde{SPS}}_{1}\)

> g1 = a4⌃2-4*a5*a3;

> g2 = b1*a4-2*b2*a3+2*b3*a2-b4*a1;

> g3 = b3*a4-b4*a3;

> g4 = b4*a4-4*b5*a3;

> g5 = d5*a4+2*c5*a3;

> g6 = d4*a4+2*c4*a3;

> g7 = d3*a4+2*c3*a3;

> g8 = 2*b1*a5-b2*a4+b4*a2-2*b5*a1;

> g9 = b3*a5-b5*a3;

> g10 = b4*a5-b5*a4;

> g11 = 2*d5*a5+c5*a4;

> g12 = 2*d4*a5+c4*a4;

> g13 = 2*d3*a5+c3*a4;

> g14 = 2*d5*b2+2*c5*b1+d4*a2+c4*a1;

> g15 = d4*b2+c4*b1+2*d3*a2+2*c3*a1;

> g16 = 2*d5*b3+d4*a3;

> g17 = d4*b3+2*d3*a3;

> g18 = 2*c5*b3+c4*a3;

> g19 = c4*b3-d3*a4;

> g20 = b4⌃2-4*b5*b3;

> g21 = d5*b4-c4*a3;

> g22 = d4*b4-4*c3*a3;

> g23 = d3*b4+2*c3*b3;

> g24 = 2*c5*b4+c4*a4;

> g25 = c4*b4+2*c3*a4;

> g26 = 4*d5*b5-c4*a4;

> g27 = d4*b5-c3*a4;

> g28 = 2*d3*b5+c3*b4;

> g29 = 2*c5*b5+c4*a5;

> g30 = c4*b5+2*c3*a5;

> g31 = d4⌃2-4*d3*d5;

> g32 = c5*d4-c4*d5;

> g33 = c4*d4-4*c3*d5;

> g34 = c5*d3-c3*d5;

> g35 = c4*d3-c3*d4;

> g36 = c4⌃2-4*c3*c5;

> g37 = 2*b3*b2*a3-b4*b1*a3-2*b3⌃2*a2+b4*b3*a1;

> g38 = b4*b2*a3-2*b5*b1*a3-b4*b3*a2+2*b5*b3*a1;

> g39 = d3*b2*a3+c3*b1*a3-d3*b3*a2-c3*b3*a1;

> g40 = d5*b1⌃2*a2+c5*b1⌃2*a1+d4*b1*a2*a1

+c4*b1*a1⌃2+d3*a2*a1⌃2+c3*a1⌃3+a3;

> g41 = d4*b1⌃2*a2+c4*b1⌃2*a1+4*d3*b1*a2*a1

-2*d3*b2*a1⌃2+2*c3*b1*a1⌃2-2*b3;

> g42 = 2*c5*b1⌃2*a2-2*c5*b2*b1*a1+c4*b1*a2*a1

-c4*b2*a1⌃2-a4;

> g43 = c4*b1⌃2*a2-c4*b2*b1*a1+2*c3*b1*a2*a1

-2*c3*b2*a1⌃2+b4;

> g44 = 2*c5*b2*b1*a2+c4*b1*a2⌃2-2*c5*b2⌃2*a1

-c4*b2*a2*a1-2*a5;

> g45 = c4*b2*b1*a2+2*c3*b1*a2⌃2-c4*b2⌃2*a1-2*c3*b2*

a2*a1+2*b5;

> g46 = 2*d3*b1⌃2*a2⌃2-4*d3*b2*b1*a2*a1

+2*d3*b2⌃2*a1⌃2+2*b3*b2-b4*b1;

> g47 = 2*c3*b1⌃2*a2⌃2-4*c3*b2*b1*a2*a1

+2*c3*b2⌃2*a1⌃2-b4*b2+2*b5*b1;

> g48 = d3*b1⌃2*a3*a2+c3*b1⌃2*a3*a1-2*d3*b3*b1*a2*a1

+d3*b3*b2*a1⌃2

-c3*b3*b1*a1⌃2+b3⌃2;

> g49 = 2*c3*b1⌃2*a3*a2-2*c3*b2*b1*a3*a1-2*c3*b3*b1*a2*a1

+2*c3*b3*b2*a1⌃2

-b4*b3;

> g50 = c3*b2*b1*a3*a2-c3*b3*b1*a2⌃2-c3*b2⌃2*a3*a1

+c3*b3*b2*a2*a1-b5*b3;

> g51 = 2*c5*b2⌃2*a3*a2+2*c4*b2*a3*a2⌃2+2*c3*a3*a2⌃3

-c5*b2⌃2*a4*a1

-c4*b2*a4*a2*a1-c3*a4*a2⌃2*a1-a5*a4;

> g52 = 2*c4*b2⌃2*a3*a2+8*c3*b2*a3*a2⌃2

-4*c3*b3*a2⌃3-c4*b2⌃2*a4*a1

-4*c3*b2*a4*a2*a1+2*c3*b4*a2⌃2*a1+2*b5*a4;

> g53 = 2*c3*b2⌃2*a3*a2-2*c3*b3*b2*a2⌃2

-c3*b4*b1*a2⌃2-c3*b2⌃2*a4*a1

+2*c3*b4*b2*a2*a1-b5*b4;

> g54 = c5*b2⌃2*a4*a2+c4*b2*a4*a2⌃2

+c3*a4*a2⌃3-2*c5*b2⌃2*a5*a1

-2*c4*b2*a5*a2*a1-2*c3*a5*a2⌃2*a1-2*a5⌃2;

> g55 = c4*b2⌃2*a4*a2+4*c3*b2*a4*a2

⌃2-2*c3*b4*a2⌃3-2*c4*b2⌃2*a5*a1

-8*c3*b2*a5*a2*a1+4*c3*b5*a2⌃2*a1+4*b5*a5;

> g56 = c3*b2⌃2*a4*a2-c3*b4*b2*a2

⌃2-2*c3*b5*b1*a2⌃2-2*c3*b2⌃2*a5*a1

+4*c3*b5*b2*a2*a1-2*b5⌃2;

> g57 = a3;

> g58 = a5;

> g59 = b3;

> g60 = 1-u*b5;

> ideal I1=g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,

g14,g15,g16,g17,g18,g19,g20,g21,g22,g23,g24,g25,g26,g27,

g28,g29,g30,g31,g32,g33,g34,g35,g36,g37,g38,g39,g40,g41,

g42,g43,g44,g45,g46,g47,g48,g49,g50,g51,g52,g53,g54,

g55,g56,g57,g58,g59,g60;

> // eliminate variables u

> ideal I11=eliminate(I1,u);

> // find the minimal irreducible decomposition for I11

> minAssGTZ(I11);

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Yu, Z., Li, L. & Liu, L. Topological Classifications for a Class of 2-dimensional Quadratic Mappings and an Application to Iterative Roots. Qual. Theory Dyn. Syst. 20, 2 (2021). https://doi.org/10.1007/s12346-020-00444-8

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