ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS

Xinjie Qian; Jiazhong Yang

doi:10.11948/20220061

2022 Volume 12 Issue 6

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Xinjie Qian, Jiazhong Yang. ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2541-2554. doi: 10.11948/20220061

Citation:

Xinjie Qian, Jiazhong Yang. ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2541-2554. doi: 10.11948/20220061

ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS

Xinjie Qian^1, ,,
Jiazhong Yang²

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: X.Qian is supported by Doctor of Entrepreneurship and Innovation Funds of Jiangsu (No. JSSCBS20210617), and PhD research startup foundation of Jinling Institute of Technology (Nos. jit-b-202049, jit-fhxm-202116). The authors are supported by National Science Foundation of China (No. 12071006)

Abstract

Abstract

In this paper, a systematic algorithm is provided to determine the sharp upper bound on the number of nontrivial rational solutions for 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. As an application, we present a thorough study for an important case, $ (m,n)=(4,9 $).
- Abel differential equations /
- nontrivial rational solution /
- Liénard system /
- rational invariant curve
MSC: 34A05, 34C05, 37C10

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References

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