On the nonexistence of constants of derivations: the proof of a theorem of Jouanolou and its development
A well-known theorem of \textit{J.-P. Jouanolou} [Équations de Pfaff algébriques. Lect. Notes Math. 708 , Springer (1979; Zbl 0477.58002)] states the following: Let \(s \geq 2\) be a natural integer, and let \(d : \mathbb C [x,y,z] \to \mathbb C [x,y,z]\) be the \(\mathbb C\)-derivation of the polynomial ring in three indeterminates defined by \(d(x) = z^s\), \(d(y) = x^s\), \(d(z) = y^s\). Then, for every \(P \in \mathbb C [x,y,z]\), the equation \(d(F) = PF\) does not admit a nontrivial solution in \(\mathbb C [x,y,z]\). In this extremely well-written paper the authors give a new and very elementary proof, using from algebraic geometry only Bézout's theorem for pairs of projective plane curves. In Jouanolou's book, there are two proofs of the theorem stated above, one by Jouanolou and the other by the reviewer of the book; the use of Bézout's theorem stems from this proof. Still other proofs were given by \textit{D. Cerveau} and \textit{A. Lins-Neto} [Ann. Inst. Fourier 41, No. 4, 883--903 (1991; Zbl 0757.34006)], \textit{A. Lins-Neto} [in: Holomorphic dynamics, Lect. Notes Math. 1345, 193--232 (1988; Zbl 0677.58036)] and \textit{H. Żoladek} [On algebraic solutions of algebraic Pfaff equations, Studia Math. 114, No. 2, 117--126 (1995; Zbl 0826.34003)]. The method of proof of Jouanolou's theorem is applied in Sections 4--6 to three multidimensional examples of differential equations which do not admit nontrivial solutions. In Section 7 it is shown that the results remain true if \(\mathbb C\) is replaced by an integral domain which contains the integers.
derivation of the polynomial ring in three indeterminates
nonexistence of constants of derivations
scientific article; zbMATH DE number 792810