Darboux theory of integrability for polynomial vector fields in taking into account the multiplicity at infinity
Darboux's theory of integrability provides a relation between the existence of invariant algebraic hypersurfaces of polynomial vector fields defined in \({\mathbb R}^n\) or \({\mathbb C}^n\), \(n\geq2,\) and the existence of first integrals. In a recent paper [J. Differ. Equations 246, No.~2, 541--551 (2009; Zbl 1163.34003)] the same authors improved the classical results taking into account the algebraic multiplicity of invariant algebraic hypersurfaces. In this paper, they extend these results by introducing the algebraic multiplicity of the hyperplane at infinity. Their main result is: Assume that a polynomial vector field \(X\) of degree \(d>0\) in \({\mathbb R}^n\) has \(p\) irreducible invariant hypersurfaces, \(f_1=0,f_2=0,\dots,f_p=0\), and an invariant hyperplane at infinity. Then: (i) If one of these \(p+1\) hypersurfaces has no defined algebraic multiplicity, then \(X\) has a rational first integral; (ii) otherwise, let \(q_i\) be the algebraic multiplicity of \(f_i\) and let \(k\) be the algebraic multiplicity of the hyperplane at infinity. If moreover, when one of these algebraic multiplicities is larger than 1, \(X\) restricted to the corresponding hypersurface has no rational first integral, then: {\parindent7mm \begin{itemize}\item[(a)] If \(\sum_{i=1}^p q_i+k\geq N+2\), then \(X\) has a real Darboux first integral, where \(N={n+d-1 \choose n}\), \item[(b)] If \(\sum_{i=1}^p q_i+k\geq N+n+1\), then \(X\) has a real rational first integral. \end{itemize}} In the paper, the algebraic multiplicity of an algebraic hypersurface \(f=0\) of degree \(m\) is introduced as the multiplicity \(f\) as a factor of the \(m\)-th extactic polynomial associated to \(X\). If this polynomial is identically zero, then it is said that \(f\) has no defined algebraic multiplicity. It is also shown, through some examples, that the hypothesis on the non-existence of a rational first integral, when some hypersuface has algebraic multiplicity greater than 1, is necessary in the theorem. Finally, the difficulties for extending their result to vector fields defined in \({\mathbb C}^n\) are discussed.
polynomial vector field in projective space
scientific article; zbMATH DE number 5641311
scientific article