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Rational constants of monomial derivations. (English) Zbl 1119.13021

Let \(k(X)=k(X_{1}, \ldots, X_{n})\) be a field of rational functions over a field \(k\) of characteristic \(0\). A monomial derivation of \(k(X)\) is a derivation \(d:k(X) \rightarrow k(X)\) such that \(d(x_{i})=x^{\beta_{i1}}_{1}\cdots x^{\beta_{in}}_{n}\), for \(i=1,\ldots,n\), where \(\beta_{ij}\) are integer. The field of constants of a monomial derivation \(d\) of \(k(X)\), that is, \(k(X)^{d}=\{ \varphi \in k(X) : d(\varphi)=0 \}\) is trivial if \(k(X)^{d}=k\). The authors present some general properties of the field of constants of monomial derivations of \(k(X)\).
The main result of this paper is a complete description of all monomial derivations of \(k(x,y,z)\), with trivial field of constants. In this description is very useful the characterization of the Votka-Volterra derivations with strict Darboux polynomials of J. Moulin Ollagnier [Qual. Theory Dyn. Syst. 2, No. 2, 307–358 (2002; Zbl 1084.34001)]. An explicit description of all monomial derivations of monic monomials of the same degree \(2,3\) and \(4\), with trivial rational constants is given in this paper.

MSC:

13N15 Derivations and commutative rings
12H05 Differential algebra
16W25 Derivations, actions of Lie algebras
34C14 Symmetries, invariants of ordinary differential equations

Citations:

Zbl 1084.34001
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References:

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