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Kernels of derivations of polynomial rings and Casimir elements

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Ukrainian Mathematical Journal Aims and scope

We propose an algorithm for the evaluation of elements of the kernel of an arbitrary derivation of a polynomial ring. The algorithm is based on an analog of the well-known Casimir element of a finite-dimensional Lie algebra. By using this algorithm, we compute the kernels of Weitzenböck derivation d(x i ) = x i−1, d(x 0) = 0, i = 0,…, n, for the cases where n ≤ 6.

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References

  1. A. Nowicki, Polynomial Derivation and Their Ring of Constants, Copernicus University, Torun (1994).

    Google Scholar 

  2. A. Van Den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhäuser, Basel (2000).

    MATH  Google Scholar 

  3. G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Springer, Berlin (2006).

    MATH  Google Scholar 

  4. D. Hilbert, Theory of Algebraic Invariants, Cambridge University, Cambridge (1993).

    MATH  Google Scholar 

  5. R. Weitzenböck, “Über die Invarianten von linearen Gruppen,” Acta Math., 58, 231–293 (1932).

    Article  MATH  MathSciNet  Google Scholar 

  6. G.-M. Greuel and G. Pfister, “Geometric quotients of unipotent group actions,” Proc. London Math. Soc., 67, No. 1, 75–105 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Cerezo, Tables des Invariants Algébriqueset Rationnels d’une Matrice Nilpotente de Petite Dimension, Prépublications Mathématiques, Université de Nice, 146 (1987).

  8. L. Bedratyuk, “A complete minimal system of covariants for the binary form of degree 7,” J. Symbol Comput., 44, No. 2, 211–220 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  9. L. P. Bedratyuk and S. L. Bedratyuk, “A complete system of covariants of the binary form of degree 8,” Mat. Visn. NTSh, 5, 11–22 (2008).

    MATH  Google Scholar 

  10. L. P. Bedratyuk, “Casimir elements of derivations of a polynomial ring,” Mat. Stud., 27, 115–119 (2007).

    MATH  MathSciNet  Google Scholar 

  11. V. V. Bavula and T. H. Lenagan, “Quadratic and cubic invariants of unipotent affine automorphisms,” J. Algebra, 320, No. 12, 4132–4155 (2008).

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Khmel’nyst’kyi National University, Khmel’nyst’kyi, Ukraine

    L. P. Bedratyuk

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  1. L. P. Bedratyuk
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 4, pp. 435–452, April, 2010.

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Bedratyuk, L.P. Kernels of derivations of polynomial rings and Casimir elements. Ukr Math J 62, 495–517 (2010). https://doi.org/10.1007/s11253-010-0367-x

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  • DOI: https://doi.org/10.1007/s11253-010-0367-x

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