Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and 𝒢ₐ-actions. II

van den Essen, Arno

doi:10.1090/S0002-9939-1994-1185282-0

Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and $\mathcal {G}_a$-actions. II
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Proc. Amer. Math. Soc. 121 (1994), 667-678 Request permission

Abstract:

This paper describes several applications of locally finite and locally nilpotent derivations; we recover the main results from the theory of polynomial flows, give a new inversion formula for polynomial automorphisms, reformulate the Eulerian conjecture for ordinary systems of differential equations in terms of locally nilpotent derivations, and give an algorithm to compute the invariant ring of a ${\mathcal {G}_a}$-action on affine n-space (if this ring is finitely generated).

References

Kossivi Adjamagbo, On polynomiality or rationality of formal power series and Jacobian conjecture, Bull. Soc. Math. Belg. Sér. A 43 (1991), no. 1-2, 9–14. Contact Franco-Belge en Algèbre (Antwerp, 1990). MR 1315766
Kossivi Adjamagbo and Arno van den Essen, Eulerian systems of partial differential equations and the Jacobian conjecture, J. Pure Appl. Algebra 74 (1991), no. 1, 1–15. MR 1129126, DOI 10.1016/0022-4049(91)90045-4
Hyman Bass, A nontriangular action of $\textbf {G}_{a}$ on $\textbf {A}^{3}$, J. Pure Appl. Algebra 33 (1984), no. 1, 1–5. MR 750225, DOI 10.1016/0022-4049(84)90019-7
Hyman Bass and Gary Meisters, Polynomial flows in the plane, Adv. in Math. 55 (1985), no. 2, 173–208. MR 772614, DOI 10.1016/0001-8708(85)90020-9
Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. (2) 121 (1985), no. 2, 283–318 (French). MR 786350, DOI 10.2307/1971174

Gröbner bases

An algorithmic method in polynomial ideal theory

Polynomial flows, symmetry groups, and conditions sufficient for injectivity of maps

Brian A. Coomes, The Lorenz system does not have a polynomial flow, J. Differential Equations 82 (1989), no. 2, 386–407. MR 1027976, DOI 10.1016/0022-0396(89)90140-X
Brian A. Coomes, Polynomial flows on $\textbf {C}^n$, Trans. Amer. Math. Soc. 320 (1990), no. 2, 493–506. MR 998353, DOI 10.1090/S0002-9947-1990-0998353-7
Brian Coomes and Victor Zurkowski, Linearization of polynomial flows and spectra of derivations, J. Dynam. Differential Equations 3 (1991), no. 1, 29–66. MR 1094723, DOI 10.1007/BF01049488
James K. Deveney and David R. Finston, Sliced $\mathbf G_a$ actions, Internat. J. Algebra Comput. 11 (2001), no. 1, 19–24. MR 1818660, DOI 10.1142/S0218196701000413
James K. Deveney and David R. Finston, Rationally triangulable automorphisms, J. Pure Appl. Algebra 72 (1991), no. 1, 1–4. MR 1115562, DOI 10.1016/0022-4049(91)90124-K
Arno van den Essen, Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms, Proc. Amer. Math. Soc. 116 (1992), no. 3, 861–871. MR 1111440, DOI 10.1090/S0002-9939-1992-1111440-5

An algorithm to compute the invariant ring of a

action on an affine variety

Hanspeter Kraft, Algebraic automorphisms of affine space, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988) Progr. Math., vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 81–105. MR 1040858
Gary H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), no. 4, 679–705. MR 683862, DOI 10.1216/RMJ-1982-12-4-679

Polynomial flows on

Gary Hosler Meisters, Inverting polynomial maps of $n$-space by solving differential equations, Delay and differential equations (Ames, IA, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 107–166. MR 1170148
Gary H. Meisters and Czesław Olech, A poly-flow formulation of the Jacobian conjecture, Bull. Polish Acad. Sci. Math. 35 (1987), no. 11-12, 725–731 (English, with Russian summary). MR 961711
Masayoshi Nagata, On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 459–462. MR 0116056
M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
Pekka Nousiainen and Moss E. Sweedler, Automorphisms of polynomial and power series rings, J. Pure Appl. Algebra 29 (1983), no. 1, 93–97. MR 704289, DOI 10.1016/0022-4049(83)90083-X
V. L. Popov, On actions of $\textbf {G}_a$ on $\textbf {A}^n$, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 237–242. MR 911143, DOI 10.1007/BFb0079241
Rudolf Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A384–A387 (French). MR 232770
David Shannon and Moss Sweedler, Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence, J. Symbolic Comput. 6 (1988), no. 2-3, 267–273. Computational aspects of commutative algebra. MR 988417, DOI 10.1016/S0747-7171(88)80047-6
Dennis M. Snow, Unipotent actions on affine space, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988) Progr. Math., vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 165–176. MR 1040863
R. Weitzenböck, Über die Invarianten von linearen Gruppen, Acta Math. 58 (1932), no. 1, 231–293 (German). MR 1555349, DOI 10.1007/BF02547779
O. Zariski, Interprétations algébrico-géométriques du quatorzième problème de Hilbert, Bull. Sci. Math. (2) 78 (1954), 155–168 (French). MR 65217