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Rational solutions and limit cycles of polynomial and trigonometric Abel equations

Luis Angel Calderon

Abstract

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of rational limit cycles when A(t), B(t) and C(t) are trigonometric polynomials with real coefficients.

Rational solutions and limit cycles of polynomial and trigonometric Abel equations

Abstract

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of rational limit cycles when A(t), B(t) and C(t) are trigonometric polynomials with real coefficients.
Paper Structure (6 sections, 16 theorems, 64 equations)

This paper contains 6 sections, 16 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Invariant curves of degree one in $x$
  3. One invariant curve
  4. Two or more invariant curves
  5. Bounds for the invariant curves in the polynomial case
  6. Darboux's Theory of Integrability

Key Result

Theorem 1.1

With the above notation and conventions, the following claims are satisfied.

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 22 more