The commuting derivations conjecture. (English) Zbl 1052.13012
The main result of the paper (theorem 3.6) is an application of Kaliman’s theorem [see e.g. S. Kaliman, Pac. J. Math. 203, No. 1, 161–190 (2002; Zbl 1060.14085)] to pairs of commuting locally nilpotent derivations of \({\mathbb C}[x,y,z]\), or equivalently, to algebraic actions of the vector group \(({\mathbb C}^2,+)\) on \({\mathbb C}^3\). The quotient morphism for such an action is of the form \(\pi :{\mathbb C}^3\to {\mathbb C}^1\), which may be viewed as a regular function on \({\mathbb C}^3\). The author shows that the general fiber of \(\pi\) is a plane. In this case, Kaliman’s theorem implies that \(\pi\) is a coordinate function on \({\mathbb C}^3\).
Reviewer’s remark: There exist algebraic actions of \({\mathbb C}^2\) on \({\mathbb C}^3\) which are not linear in any coordinate system. However, the results of this paper show that any such action has rank at most 2. From this, it is possible to classify all \({\mathbb C}^2\)-actions on \({\mathbb C}^3\), though the paper under review does not follow this course: see D. Daigle and G. Freudenburg, J. Algebra 204, No.2, 353-371 (1998; Zbl 0956.13007).
Reviewer’s remark: There exist algebraic actions of \({\mathbb C}^2\) on \({\mathbb C}^3\) which are not linear in any coordinate system. However, the results of this paper show that any such action has rank at most 2. From this, it is possible to classify all \({\mathbb C}^2\)-actions on \({\mathbb C}^3\), though the paper under review does not follow this course: see D. Daigle and G. Freudenburg, J. Algebra 204, No.2, 353-371 (1998; Zbl 0956.13007).
Reviewer: Gene Freudenburg (Evansville)
MSC:
| 13N15 | Derivations and commutative rings |
| 13B25 | Polynomials over commutative rings |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
References:
| [1] | Abhyankar, S.; Moh, T., Embeddings of the line in the plane, J. Reine Angew. Math., 276, 148-166 (1975) · Zbl 0332.14004 |
| [2] | H. Derksen, A. van den Essen, P. van Rossum, The cancellation problem in dimension four, Report No. 0022, University of Nijmegen, 2000.; H. Derksen, A. van den Essen, P. van Rossum, The cancellation problem in dimension four, Report No. 0022, University of Nijmegen, 2000. |
| [3] | E. Edo, S. Vénéreau, Length 2 variables of \(AXY\); E. Edo, S. Vénéreau, Length 2 variables of \(AXY\) |
| [4] | A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, in: Progress of Mathematics, Vol. 190, Birkhäuser, Basel, 2000.; A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, in: Progress of Mathematics, Vol. 190, Birkhäuser, Basel, 2000. · Zbl 0962.14037 |
| [5] | A. van den Essen, P. van Rossum, Triangular derivations related to problems on affine \(n\); A. van den Essen, P. van Rossum, Triangular derivations related to problems on affine \(n\) · Zbl 0995.14023 |
| [6] | Fujita, T., On Zariski problem, Proc. Japan Acad. Ser. A, Math. Sci., 55, 106-110 (1979) · Zbl 0444.14026 |
| [7] | S. Kaliman, Polynomials with general ((C^2\); S. Kaliman, Polynomials with general ((C^2\) |
| [8] | Kaliman, S.; Koras, M.; Makar-Limanov, L.; Russell, P., \(C^*-actions on C^3\) are linearizable, Electron. Res. Announcements AMS, 3, 63-71 (1997) · Zbl 0890.14026 |
| [9] | S. Kaliman, S. Vénéreau, M. Zaidenberg, Simple birational extensions of the polynomial ring ((C^{[3]}\); S. Kaliman, S. Vénéreau, M. Zaidenberg, Simple birational extensions of the polynomial ring ((C^{[3]}\) |
| [10] | H. Kraft, Challenging problems on affine \(n\); H. Kraft, Challenging problems on affine \(n\) |
| [11] | Makar-Limanov, L., On the hypersurface \(x+x^2y+z^2+t^3=0\) in \(C^4 or a C^3\)-like threefold which is not \(C^3\), Israel J. Math., 96, 419-429 (1996) · Zbl 0896.14021 |
| [12] | M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological theories— to the memory of Dr. Takehiko Miyata, Kinokuniya, Tokyo, 1995, pp. 37-51.; M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological theories— to the memory of Dr. Takehiko Miyata, Kinokuniya, Tokyo, 1995, pp. 37-51. |
| [13] | M. Myanishi, Curves on rational and unirational surfaces, Tata Institute of Fundamental Research 60, Tata Institute, 1978.; M. Myanishi, Curves on rational and unirational surfaces, Tata Institute of Fundamental Research 60, Tata Institute, 1978. · Zbl 0425.14008 |
| [14] | Rentschler, R., Opérations du groupe additif sur le plan, C.R.Acad. Sci. Paris, 267, 384-387 (1968) · Zbl 0165.05402 |
| [15] | P. van Rossum, Tackling problems on affine space with locally nilpotent derivations on polynomial rings, Ph.D. Thesis, University of Nijmegen, 2001.; P. van Rossum, Tackling problems on affine space with locally nilpotent derivations on polynomial rings, Ph.D. Thesis, University of Nijmegen, 2001. |
| [16] | Russell, P., Simple birational extensions of two dimensional affine rational domains, Compositio Math., 33, 197-208 (1976) · Zbl 0342.13003 |
| [17] | Sathaye, A., On linear planes, Proc. Amer. Math. Soc., 56, 1-7 (1976) · Zbl 0345.14013 |
| [18] | Wright, D., Cancellation of variables of the form \(bT^n \)−\(a\), J. Algebra, 52, 94-100 (1978) · Zbl 0383.13014 |
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