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On the classification of simple quadratic derivations over the affine plane. (English) Zbl 1152.13019
Let \(R\) be a commutative ring with \(1\). A derivation \(d: R \to R\) is simple if there are no non-zero proper ideals \(I \subseteq R\) such that \(d(I) \subseteq I\). In this paper, the author studies the simple derivations of \(R\) when \(R = \mathbb{C}[x,y]\).
Let \(d_u = a \, \partial/\partial x + b \, \partial/\partial y\) be the derivation of \(R = \mathbb{C}[x,y]\) associated with the pair \(u = (a,b)\), where \(a,b \in R\). For \(d\) to be a simple derivation, it is necessary that \(u\) is a unimodular row, i.e. that the ideal \((a,b) = R\). Denote by \(U_n\) the set of unimodular rows \(u = (a,b)\) with \(\max \{ \deg a, \deg b \} = n\), and let \(\overline{U_2}\) be the closure of \(U_2\) in \(R_2 \times R_2\). The author gives an explicit description of the four irreducible components \(\{ P_j: j = 1,2,3,4 \}\) of \(\overline{U_2}\). In particular, \(\dim P_j = 8\) for \(j = 1,2,3\) and \(\dim P_4 = 7\), and \(U_2 \cap P_j\) is dense in \(P_j\) for all \(j\).
Let \(\Delta_j \subseteq U_2 \cap P_j\) be the set of unimodular rows \(u \in U_2 \cap P_j\) such that \(d_u\) is a simple derivation. The author shows that \(\Delta_j\) is non-empty for all \(j\), and dense in \(P_j\) for \(j = 1,2\). It is an open question if \(\Delta_j\) is dense in \(P_j\) for \(j = 3,4\). The author has used computer algebra methods to obtain these results, but has managed to avoid the use of computer algebra arguments in most proofs. Several new families of simple derivations are given as part of the proof.

13N15 Derivations and commutative rings
Full Text: DOI
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