×

Search for proper polynomials of the differential operator \(P{\partial \over {\partial x}}+Q{\partial\over {\partial y}}\) over \(\mathbb{C}[x,y]\). (English. Russian original) Zbl 0792.12002

Mosc. Univ. Math. Bull. 48, No. 2, 24-27 (1993); translation from Vestn. Mosk. Univ., Ser. I 48, No. 2, 27-31 (1993).
Let \(R\) be a domain of integrity, \(d\) the differentiation on \(R\) and \(F\) a field of quotients of \(R\). Denote \(\ell\delta\) the logarithmic derivative on \(F\) defined by \(\ell\delta(f)= f^{-1} df\). The author states the problem of describing all irreducible \(p\in R\) such that \(\ell\delta(p)\in R\). He shows that the use of formal power series can bring a complete solution to the problem in some cases.

MSC:

12H05 Differential algebra