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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.

MSC:
 13N15 Derivations and commutative rings
derivation
SINGULAR
Full Text:
References:
 [1] Artin, M., Algebra, (1991), Prentice Hall · Zbl 0788.00001 [2] Carnicer, M.N., The Poincaré problem in the nondicritical case, Ann. of math., 140, 289-294, (1994) · Zbl 0821.32026 [3] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, (1992), Springer-Verlag · Zbl 0756.13017 [4] Coutinho, S.C., d-simple rings and simple D-modules, Math. proc. Cambridge philos. soc., 125, 405-415, (1999) · Zbl 0935.16017 [5] Coutinho, S.C.; Menasché Schechter, L., Algebraic solutions of holomorphic foliations: an algorithmic approach, J. symbolic comput., 41, 5, 603-618, (2006) · Zbl 1134.32011 [6] Daly, T., \scaxiom: the thirty year horizon, vol. 1: tutorial, (2005), Lulu Press [7] Doering, A.M.; Lequain, Y.; Ripoll, C.C., Differential simplicity and cyclic maximal left ideals of the Weyl algebra $$A_2(K)$$, Glasgow math. J., 48, 269-274, (2006) · Zbl 1104.16019 [8] Goodearl, K.R.; Warfield, R.B., An introduction to noncommutative Noetherian rings, London math. soc. stud. texts, vol. 16, (1989), Cambridge University Press · Zbl 0679.16001 [9] Humphreys, J.E., Linear algebraic groups, (1981), Springer-Verlag · Zbl 0507.20017 [10] Greuel, G.-M.; Pfister, G.; Schönemann, H., \scsingular 3.0. A computer algebra system for polynomial computations, (2005), Centre for Computer Algebra, University of Kaiserslautern [11] Jordan, D., Differentiably simple rings with no invertible derivatives, Q. J. math. Oxford (2), 32, 417-424, (1981) · Zbl 0471.13014 [12] Man, Y.-K.; MacCallum, M.A.H., A rational approach to the prelle – singer algorithm, J. symbolic comput., 24, 31-43, (1997) · Zbl 0922.12007 [13] Mumford, D., The red book of varieties and schemes, (1999), Springer-Verlag [14] Maciejewski, A.; Moulin Ollagnier, J.; Nowicki, A., Simple quadratic derivations in two variables, Comm. algebra, 29, 5095-5113, (2001) · Zbl 1014.13007 [15] Shafarevich, I.R., Basic algebraic geometry, (1977), Springer-Verlag Berlin/Heidelberg · Zbl 0362.14001 [16] Walcher, S., On the Poincaré problem, J. differential equations, 166, 51-78, (2000) · Zbl 0974.34028 [17] Zakeri, S., Dynamics of singular holomorphic foliations on the complex projective plane, (), 179-233 · Zbl 1193.37066
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