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Simple quadratic derivations in two variables. (English) Zbl 1014.13007
A derivation \(d\) of a commutative \(k\)-algebra \(R\) is said to be simple if \(R\) has no \(d\)-invariant ideals other than \(0\) and \(R\). Here, \(k\) is a characteristic zero field. The authors study quadratic derivations of the bivariate polynomial ring \(k[x,y]\), where \(k\) is any field of characteristic 0, and investigate which of these are simple. Specifically, quadratic derivations are those of the form \((\partial_x+q\partial_y)\), where \(q\) is degree-2 monic in y with coefficients in \(k[x]\). By changing coordinates, one may assume \(q=y^2+p(x)\), and the derivation is then labeled \(\Delta_p\).
The main result (theorem 6.1) is that, if \(\Delta_p\) is not simple, there exists an invariant principal ideal generated by a polynomial of \(y\)-degree one. This allows the authors to give several specific classes of simple derivations of the form \(\Delta_p\) on \(k[x,y]\).

13N15 Derivations and commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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