About a conjecture on quadratic vector fields
Let \(U\) be a partial differential field of characteristic 0 with differentiations \(\delta_{1},\dots ,\delta_{m}\) and \(R\) be a factorial differential subring of \(U\) with a field of quotients \(F\). Denote by \(C\) a field of constants of \(F\). For any \(r_{1},\dots ,r_{m} \in R, \delta=r_{1}\delta_{1}+\cdots +r_{m}\delta_{m}\) is a differentiation on \(R\). The differentiation \(\delta\) is called full if from \(\delta f = 0 \Rightarrow f\in C\), \(\forall f\in F\). An element \(f\in F\) is called a rational first integral of degree 0 (with respect to \(\delta\)) if \(f\notin C\) and \(\delta f=0\). An element \(t\in R\setminus C\) is called a Darboux element if \(\delta t=\lambda t\) for some \(\lambda\in R\). The answer to the question whether the given differentiation is full or not full is not trivial even if \(m=2\) [see \textit{A. V. Astrelin}, Mosc. Univ. Math. Bull. 48, No. 2, 24--27 (1993); translation from Vestn. Mosk. Univ., Ser. I 48, 27--31 (1993; Zbl 0792.12002)]. If \(R\) does not contain Darboux elements then it is differentially integrally closed (with respect to \(\delta\)) in \(F\) [see \textit{J. Brzezinski}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 10, 325--328 (1962; Zbl 0104.26301) and \textit{H. Nishimura}, Proc. Japan Acad. 40, 145--149 (1964; Zbl 0143.06101)]. The existence of the first integrals for \(\delta\) deprives \(R\) of the above property and, hence, causes the presence of Darboux elements. The opposite is incorrect, as the author shows in this paper. He considers the case where NEWLINE\[NEWLINER=C[x, y, z],\;\delta_{1}=\frac{\partial}{\partial x},\;\delta_{2}=\frac{\partial}{\partial y},\;\delta_{3}=\frac{\partial}{\partial z}NEWLINE\]NEWLINE and provides the whole family of full (Lotka-Volterra) differentiations of \(R\), which have Darboux elements.
scientific article; zbMATH DE number 1698496
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