THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS

Xinjie Qian; Yang Shen; Jiazhong Yang

doi:10.11948/20200475

2021 Volume 11 Issue 5

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Xinjie Qian, Yang Shen, Jiazhong Yang. THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2535-2552. doi: 10.11948/20200475

Citation:

Xinjie Qian, Yang Shen, Jiazhong Yang. THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2535-2552. doi: 10.11948/20200475

THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS

1.
College of Science, Jinling Institute of Technology, Nanjing, 211169, China
2.
School of Mathematical Sciences, Peking University, Beijing, 100871, China

Corresponding author: Email: qianxj@jit.edu.cn(X. Qian)
Fund Project: The authors were supported by National Science Foundation of China (No. NSFC 12071006), and Qian is supported by PhD research startup foundation of Jinling Institute of Technology (No. jit-b-202049)

Abstract

Abstract

In this paper, we study rational solutions of the Abel differential equations $ dy/dx = f_m(x)y^2+g_n(x)y^3 $, where $ f_m(x) $ and $ g_n(x) $ are real polynomials of degree $ m $ and $ n $ respectively. The main result of the paper is as follows: We give a systematic upper bound on the number of the nontrivial rational solutions of such equations in all these cases. Then we prove that these upper bounds can be reached in most cases. Finally, we present some examples of Abel equations having exactly $ i $ nontrivial rational solutions, where $ 1\leq i\leq 5 $.
- Abel differential equations /
- nontrivial rational solution /
- Liénard system /
- rational invariant curve
MSC: 34A05, 34C05, 37C10

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References

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