Generalized Nowicki conjecture
HTML articles powered by AMS MathViewer
- by Vesselin Drensky PDF
- Proc. Amer. Math. Soc. 148 (2020), 3705-3711 Request permission
Abstract:
Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $\delta$ of $B[Y_d]=B[y_1,\ldots ,y_d]$ is elementary if $\delta (B)=0$ and $\delta (y_i)\in B$, $i=1,\ldots ,d$. Then for $d\geq 2$ the elements $u_{ij}=\delta (y_i)y_j-\delta (y_j)y_i$, $1\leq i<j\leq d$, belong to the algebra $B[Y_d]^{\delta }$ of constants of $\delta$, and it is a natural question whether the $B$-algebra $B[Y_d]^{\delta }$ is generated by all $u_{ij}$. In this paper we consider the special case of $B=K[X_d]=K[x_1,\ldots ,x_d]$. If $\delta (y_i)=x_i$, $i=1,\ldots ,d$, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case $\delta (y_i)=x_i^{n_i}$, $n_i>0$, $i=1,\ldots ,d$, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if $\delta (y_i)=f_i(x_i)$, for any nonconstant polynomials $f_i(x_i)$, $i=1,\ldots ,d$, then $B[Y_d]^{\delta }=K[X_d,Y_d]^{\delta }$ is generated by $X_d$ and $U_d=\{u_{ij}=f_i(x_i)y_j-y_if_j(x_j)\mid 1\leq i<j\leq d\}$. In the present paper we have found a presentation of the algebra \[ K[X_d,Y_d]^{\delta }=K[X_d,U_d\mid R=S=0],d\geq 4, \] \[ R=\{r(i,j,k,l)\mid 1\leq i<j<k<l\leq d\}, S=\{s(i,j,k)\mid 1\leq i<j<k\leq d\}, \] and a basis of $K[X_d,Y_d]^{\delta }$ as a vector space. As a corollary we have shown that the defining relations $R\cup S$ form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras $K[X_d,Y_d]^{\delta }$, $d<4$, are also described.References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
- Vesselin Drensky, Another proof of the Nowicki conjecture, Tokyo J. Math. (to appear), arXiv:1902.08758 [math.AC], 2019, DOI 10.3836/tjm/1502179320.
- Vesselin Drensky and Leonid Makar-Limanov, The conjecture of Nowicki on Weitzenböck derivations of polynomial algebras, J. Algebra Appl. 8 (2009), no. 1, 41–51. MR 2191532, DOI 10.1142/S0219498809003217
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Gene Freudenburg, A survey of counterexamples to Hilbert’s fourteenth problem, Serdica Math. J. 27 (2001), no. 3, 171–192. MR 1917641
- Joseph Khoury, Locally nilpotent derivations and their rings of constants, Ph.D. Thesis, University of Ottawa, 2001.
- Joseph Khoury, A note on elementary derivations, Serdica Math. J. 30 (2004), no. 4, 549–570. MR 2110495
- Joseph Khoury, A Groebner basis approach to solve a conjecture of Nowicki, J. Symbolic Comput. 43 (2008), no. 12, 908–922. MR 2472540, DOI 10.1016/j.jsc.2008.05.004
- Shigeru Kuroda, A simple proof of Nowicki’s conjecture on the kernel of an elementary derivation, Tokyo J. Math. 32 (2009), no. 1, 247–251. MR 2541159, DOI 10.3836/tjm/1249648420
- Andrzej Nowicki, Polynomial derivations and their rings of constants, Uniwersytet Mikołaja Kopernika, Toruń, 1994. MR 2553232
- Andrzej Nowicki, The fourteenth problem of Hilbert for polynomial derivations, Differential Galois theory (Będlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 177–188. MR 1972453, DOI 10.4064/bc58-0-13
- R. Weitzenböck, Über die Invarianten von linearen Gruppen, Acta Math. 58 (1932), no. 1, 231–293 (German). MR 1555349, DOI 10.1007/BF02547779
Additional Information
- Vesselin Drensky
- Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
- MR Author ID: 59730
- Email: drensky@math.bas.bg
- Received by editor(s): March 5, 2019
- Received by editor(s) in revised form: November 20, 2019
- Published electronically: June 4, 2020
- Additional Notes: The author was partially supported by Grant KP-06-N-32/1 of the Bulgarian National Science Fund.
- Communicated by: Jerzy Weyman
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3705-3711
- MSC (2010): Primary 13N15, 13P10; Secondary 13E15
- DOI: https://doi.org/10.1090/proc/15068
- MathSciNet review: 4127818