Article
| Title: | Remarks on retracts of polynomial rings in three variables in any characteristic (English) |
| Author: | Kojima, Hideo |
| Author: | Nagamine, Takanori |
| Author: | Sasagawa, Riko |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1197-1211 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $A$ be a retract of the polynomial ring in three variables over a field $k$. It is known that if ${\rm char} (k) = 0$ or ${\rm tr.deg}_k A \not = 2$, then $A$ is a polynomial ring. We give some sufficient conditions for $A$ to be the polynomial ring in two variables over $k$ when ${\rm char} (k) > 0$ and ${\rm tr.deg}_k A = 2$. (English) |
| Keyword: | retract |
| Keyword: | polynomial ring |
| Keyword: | exponential map |
| MSC: | 13A50 |
| MSC: | 13B25 |
| MSC: | 13N15 |
| MSC: | 14R20 |
| DOI: | 10.21136/CMJ.2025.0001-25 |
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| Date available: | 2025-12-20T07:19:03Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153237 |
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| Reference: | [1] Abhyankar, S. S., Heinzer, W., Eakin, P.: On the uniqueness of the coefficient ring in a polynomial ring.J. Algebra 23 (1972), 310-342. Zbl 0255.13008, MR 0306173, 10.1016/0021-8693(72)90134-2 |
| Reference: | [2] Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings.Cambridge Studies in Advanced Mathematics 144. Cambridge University Press, Cambridge (2015). Zbl 1360.14001, MR 3307753, 10.1017/CBO9781139175852 |
| Reference: | [3] Bhatwadekar, S. M., Gupta, N.: A note on the cancellation property of $k[X,Y]$.J. Algebra Appl. 14 (2015), Article ID 1540007, 5 pages. Zbl 1326.14142, MR 3368259, 10.1142/S0219498815400071 |
| Reference: | [4] Chakraborty, S., Dasgupta, N., Dutta, A. K., Gupta, N.: Some results on retracts of polynomial rings.J. Algebra 567 (2021), 243-268. Zbl 1468.13060, MR 4158731, 10.1016/j.jalgebra.2020.08.030 |
| Reference: | [5] Chakraborty, S., Gurjar, R. V., Miyanishi, M.: Factorially closed subrings of commutative rings.Algebra Number Theory 9 (2015), 1137-1158. Zbl 1318.13002, MR 3366001, 10.2140/ant.2015.9.1137 |
| Reference: | [6] Costa, D. L.: Retracts of polynomial rings.J. Algebra 44 (1977), 492-502. Zbl 0352.13008, MR 0429866, 10.1016/0021-8693(77)90197-1 |
| Reference: | [7] Evyatar, A., Zaks, A.: Rings of polynomials.Proc. Am. Math. Soc. 25 (1970), 559-562. Zbl 0198.06202, MR 0258820, 10.1090/S0002-9939-1970-0258820-3 |
| Reference: | [8] Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations.Encyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic Transformation Groups 7. Springer, Berlin (2017). Zbl 1391.13001, MR 3700208, 10.1007/978-3-662-55350-3 |
| Reference: | [9] Gupta, N.: On the cancellation problem for the affine space $\Bbb A^3$ in characteristic $p$.Invent. Math. 195 (2014), 279-288. Zbl 1309.14050, MR 3148104, 10.1007/s00222-013-0455-2 |
| Reference: | [10] Gupta, N.: On Zariski's cancellation problem in characteristic.Adv. Math. 264 (2014), 296-307. Zbl 1325.14078, MR 3250286, 10.1016/j.aim.2014.07.012 |
| Reference: | [11] Gupta, N., Nagamine, T.: Retracts of Laurent polynomial rings.Available at https://arxiv.org/abs/2301.12681 (2023), 5 pages. 10.48550/arXiv.2301.12681 |
| Reference: | [12] Kambayashi, T.: On the absence of nontrivial separable forms of the affine plane.J. Algebra 35 (1975), 449-456. Zbl 0309.14029, MR 0369380, 10.1016/0021-8693(75)90058-7 |
| Reference: | [13] Kambayashi, T.: On Fujita's cancellation theorem for the affine plane.J. Fac. Sci., Univ. Tokyo, Sect IA, Math. 27 (1980), 535-548. Zbl 0453.14015, MR 0603951 |
| Reference: | [14] Kojima, H.: Notes on the kernels of locally finite higher derivations in polynomial rings.Commun. Algebra 44 (2016), 1924-1930. Zbl 1344.13008, MR 3490655, 10.1080/00927872.2015.1027387 |
| Reference: | [15] Kojima, H.: Smooth affine $\Bbb{G}_m$-surfaces with finite Picard groups and trivial units.Tokyo J. Math. 46 (2023), 93-109. Zbl 1522.14079, MR 4609895, 10.3836/tjm/1502179385 |
| Reference: | [16] Kuroda, S.: A generalization of Nakai's theorem on locally finite iterative higher derivations.Osaka J. Math. 54 (2017), 335-341. Zbl 1368.13027, MR 3657233 |
| Reference: | [17] Liu, D., Sun, X.: A class of retracts of polynomial algebras.J. Pure Appl. Algebra 222 (2018), 382-386. Zbl 1387.14147, MR 3694460, 10.1016/j.jpaa.2017.04.009 |
| Reference: | [18] Liu, D., Sun, X.: Retracts that are kernels of locally nilpotent derivations.Czech. Math. J. 72 (2022), 191-199. Zbl 07511561, MR 4389114, 10.21136/CMJ.2021.0388-20 |
| Reference: | [19] Miyanishi, M.: Normal affine subalgebras of a polynomial ring.Algebraic and Topological Theories Kinokuniya Company, Tokyo (1986), 37-51. Zbl 0800.14018, MR 1102251 |
| Reference: | [20] Nagamine, T.: A note on retracts of polynomial rings in three variables.J. Algebra 534 (2019), 339-343. Zbl 1423.13063, MR 3979078, 10.1016/j.jalgebra.2019.05.040 |
| Reference: | [21] Russell, P., Sathaye, A.: On finding and cancelling variables in $k[X,Y,Z]$.J. Algebra 57 (1979), 151-166. Zbl 0411.13011, MR 0533106, 10.1016/0021-8693(79)90214-X |
| Reference: | [22] Shpilrain, V., Yu, J.-T.: Polynomial retracts and the Jacobian conjecture.Trans. Am. Math. Soc. 352 (2000), 477-484. Zbl 0944.13011, MR 1487631, 10.1090/S0002-9947-99-02251-5 |
| Reference: | [23] Zariski, O.: Interprétations algébrico-géométriques du quatorzième problème de Hilbert.Bull. Sci. Math., II. Sér. 78 (1954), 155-168 French. Zbl 0056.39602, MR 0065217 |
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