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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 $$k[x,y]$$.

MSC:
 13N15 Derivations and commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
Keywords:
simple derivations; polynomial rings
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