Algebraic solutions for polynomial systems with emphasis in the quadratic case
A polynomial system is a differential system of the form \[ {dx\over dt}= P(x,y), \quad {dy\over dt} =Q(x,y) \tag{1} \] where \(P\) and \(Q\) are polynomials with real or complex coefficients, i.e., \(P,Q\in\mathbb{R} [x,y]\) or \(P,Q\in \mathbb{C} [x,y]\), respectively. The number \(m= \max \{\deg P,\deg Q\}\) is called the degree of the system, and when \(m=2\), (1) is called a quadratic system. The following results on the link between polynomial systems and algebraic curves are obtained. Theorem: Assume that a real (complex) polynomial system of degree \(m\) admits \(q=[m(m+1)/2] +1-p\) real (complex) algebraic solutions \(f_i=0\), \(i=1, \dots,q\), not passing through \(p\) real (complex) independent singular points \((x_k,y_k)\), \(k=1,\dots,p\), then the system has a first integral of the form \(f_1^{\lambda_1} \dots f_q^{\lambda_q}\) with \(\lambda_i \in\mathbb{R} (\mathbb{C})\). Theorem: Assume that a real polynomial system of degree \(m\) admits \(p= [m(m+1)/2]-p\) real algebraic solutions \(f_i=0\), \(i=1, \dots, q\), not passing through \(p\) real independent weak singular points \((x_k,y_k)\), \(k=1, \dots,p\), then the system has a first integral of the form \(f_1^{\lambda_1} \dots f_q^{\lambda_q} V^\lambda\) with \(\lambda_i\)'s and \(\lambda\) real, and \(V\) being a real analytic function. In the case when (1) is a quadratic system, the following statements are established. (i) Let \(q\in \{1,2,3\}\). Assume that a real (complex) quadratic system has \(q\) real (complex) algebraic solutions \(f_i=0\), \(i=1, \dots,q\), not passing through \(4-q\) singular points, then the system either has a first integral of the form \(f_1^{\lambda_1} \dots f_q^{\lambda_q}\) with \(\lambda_i\) real (complex), or it has an elementary first integral. (ii) Let \(q\in \{0,1,2\}\). If a quadratic system has \(3-q\) real singular points, not contained in any algebraic curve which has some simple closed component, then it has at most \(q\) real algebraic curves containing limit cycles. (iii) Let \(q\in \{1,2\}\). If a real quadratic system has \(q\) real algebraic solutions, \(f_i=0\), \(i=1, \dots,q\), not passing through \(3-q\) weak singular points, then it has a first integral of the form \(f_1^{\lambda_1} \dots f_q^{\lambda_q} V^\lambda\) with \(\lambda_i\)'s and \(\lambda\) real, and \(V\) being a real analytic function. (iv) If a quadratic system has algebraic limit cycles, then it has at most one weak singular point not contained in any algebraic solution.
real (complex) independent singular points
scientific article; zbMATH DE number 1051304