Skip to main content

2017 | OriginalPaper | Buchkapitel

11. Epilogue

verfasst von : Gene Freudenburg

Erschienen in: Algebraic Theory of Locally Nilpotent Derivations

Verlag: Springer Berlin Heidelberg

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Many open questions, ranging from specific cases to broader themes, have already been posed and discussed in the foregoing chapters. A solution to the Embedding Problem or Cancellation Problem for complex affine spaces would reverberate across the whole of algebra, and we have seen how locally nilpotent derivations might play a role in their solution. Following are several additional directions for future inquiry.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
12.
Zurück zum Zitat H. Bass, A non-triangular action of \(\mathbb{G}_{a}\) on \(\mathbb{A}^{3}\), J. Pure Appl. Algebra 33 (1984), 1–5.MathSciNetCrossRef H. Bass, A non-triangular action of \(\mathbb{G}_{a}\) on \(\mathbb{A}^{3}\), J. Pure Appl. Algebra 33 (1984), 1–5.MathSciNetCrossRef
28.
Zurück zum Zitat S. Bhatwadekar, N. Gupta, and S. Lokhande, Somek-theoretic properties of the kernel of a locally nilpotent derivation onk[x 1, …, x 4], Trans. Amer. Math. Soc. 369 (2017), 341–363. S. Bhatwadekar, N. Gupta, and S. Lokhande, Somek-theoretic properties of the kernel of a locally nilpotent derivation onk[x 1, , x 4], Trans. Amer. Math. Soc. 369 (2017), 341–363.
101.
Zurück zum Zitat V. I. Danilov and M. H. Gizatullin, Fields of \(\mathbb{G}_{a}\) invariants are ruled, Canad. Math. Bull. 37 (1994), 37–41. V. I. Danilov and M. H. Gizatullin, Fields of \(\mathbb{G}_{a}\) invariants are ruled, Canad. Math. Bull. 37 (1994), 37–41.
103.
Zurück zum Zitat V. I. Danilov and M. H. Gizatullin, Algebraic aspects of additive group actions on complex affine space, Automorphisms of affine spaces (Dordrecht) (A. van den Essen, ed.), Kluwer, 1995, pp. 179–190. V. I. Danilov and M. H. Gizatullin, Algebraic aspects of additive group actions on complex affine space, Automorphisms of affine spaces (Dordrecht) (A. van den Essen, ed.), Kluwer, 1995, pp. 179–190.
128.
Zurück zum Zitat V. I. Danilov and M. H. Gizatullin, The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant, Transform. Groups 14 (2009), 531–539.MathSciNetCrossRefMATH V. I. Danilov and M. H. Gizatullin, The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant, Transform. Groups 14 (2009), 531–539.MathSciNetCrossRefMATH
132.
Zurück zum Zitat E. Dufresne and A. Maurischat, On the finite generation of additive group invariants in positive characteristic, J. Algebra 324 (2010), 1952–1963.MathSciNetCrossRefMATH E. Dufresne and A. Maurischat, On the finite generation of additive group invariants in positive characteristic, J. Algebra 324 (2010), 1952–1963.MathSciNetCrossRefMATH
149.
Zurück zum Zitat E. B. Elliott, On Weitzenböck’s theorem in positive characteristic, Proc. Amer. Math. Soc. 64 (1977), 209–213.MathSciNetMATH E. B. Elliott, On Weitzenböck’s theorem in positive characteristic, Proc. Amer. Math. Soc. 64 (1977), 209–213.MathSciNetMATH
157.
Zurück zum Zitat G. Freudenburg, Canonical factorization of the quotient morphism for an affine \(\mathbb{G}_{a}\) -variety, arXiv:1602.08786v2. G. Freudenburg, Canonical factorization of the quotient morphism for an affine \(\mathbb{G}_{a}\) -variety, arXiv:1602.08786v2.
164.
Zurück zum Zitat K.-H. Fieseler, A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven, Proc. Amer. Math. Soc. 135 (2007), 51–57.MathSciNet K.-H. Fieseler, A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven, Proc. Amer. Math. Soc. 135 (2007), 51–57.MathSciNet
171.
Zurück zum Zitat G. Freudenburg and S. Kuroda, Cable algebras and rings of \(\mathbb{G}_{a}\) -invariants, Kyoto J. Math. 57 (2017), 325–363. G. Freudenburg and S. Kuroda, Cable algebras and rings of \(\mathbb{G}_{a}\) -invariants, Kyoto J. Math. 57 (2017), 325–363.
231.
Zurück zum Zitat H. W. E. Jung, Ak-invariant of affine domains, Affine Algebraic Geometry (Osaka, Japan), Osaka University Press, 2007, pp. 231–255. H. W. E. Jung, Ak-invariant of affine domains, Affine Algebraic Geometry (Osaka, Japan), Osaka University Press, 2007, pp. 231–255.
239.
Zurück zum Zitat H. W. E. Jung, Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra 60 (1979), 439–451.MathSciNetCrossRef H. W. E. Jung, Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra 60 (1979), 439–451.MathSciNetCrossRef
260.
Zurück zum Zitat K. Kurano, Positive characteristic finite generatiion of symbolic Rees algebra and Roberts’ counterexamples to the fourteenth problem of Hilbert, Tokyo J. Math. 16 (1993), 473–496.MathSciNetCrossRefMATH K. Kurano, Positive characteristic finite generatiion of symbolic Rees algebra and Roberts’ counterexamples to the fourteenth problem of Hilbert, Tokyo J. Math. 16 (1993), 473–496.MathSciNetCrossRefMATH
306.
Zurück zum Zitat J. H. McKay and, Recent developments in affine algebraic geometry: (From the personal viewpoints of the author), Affine Algebraic Geometry, Osaka Univ. Press, Osaka, 2007, pp. 307–378. J. H. McKay and, Recent developments in affine algebraic geometry: (From the personal viewpoints of the author), Affine Algebraic Geometry, Osaka Univ. Press, Osaka, 2007, pp. 307–378.
323.
Zurück zum Zitat J. H. McKay and, Lectures on the Fourteenth Problem of Hilbert, Lecture Notes, vol. 31, Tata Inst., Bombay, 1965. J. H. McKay and, Lectures on the Fourteenth Problem of Hilbert, Lecture Notes, vol. 31, Tata Inst., Bombay, 1965.
341.
Zurück zum Zitat V. L. Popov, Bass’ triangulability problem, Adv. Stud. Pure Math., Math. Soc. Japan (to appear). V. L. Popov, Bass’ triangulability problem, Adv. Stud. Pure Math., Math. Soc. Japan (to appear).
347.
Zurück zum Zitat V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proc. Lecture Notes 54 (2011), 289–311.MathSciNetCrossRefMATH V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proc. Lecture Notes 54 (2011), 289–311.MathSciNetCrossRefMATH
348.
Zurück zum Zitat V. L. Popov, Some subgroups of the Cremona groups, Affine Algebraic Geometry (Hackensack, NJ), Lectures Notes in Math., vol. 1271, World Sci. Publ., 2013, pp. 213–242. V. L. Popov, Some subgroups of the Cremona groups, Affine Algebraic Geometry (Hackensack, NJ), Lectures Notes in Math., vol. 1271, World Sci. Publ., 2013, pp. 213–242.
380.
Zurück zum Zitat I. R. Shafarevich, On some infinite dimensional groups, Rend. Mat. Appl. (5) 25 (1966), 208–212. I. R. Shafarevich, On some infinite dimensional groups, Rend. Mat. Appl. (5) 25 (1966), 208–212.
405.
Zurück zum Zitat R. G. Swan, Representations of \(\mathbb{G}_{a}\) of codimension two, Affine Algebraic Geometry: Proceedings of the Conference, World Scientific Publishing, 2013, pp. 279–284. R. G. Swan, Representations of \(\mathbb{G}_{a}\) of codimension two, Affine Algebraic Geometry: Proceedings of the Conference, World Scientific Publishing, 2013, pp. 279–284.
428.
Zurück zum Zitat D. L. Wright, The generalized amalgamated product structure of the tame automorphism group in dimension three, Transform. Groups 20 (2014), 291–304.MathSciNetCrossRefMATH D. L. Wright, The generalized amalgamated product structure of the tame automorphism group in dimension three, Transform. Groups 20 (2014), 291–304.MathSciNetCrossRefMATH
Metadaten
Titel
Epilogue
verfasst von
Gene Freudenburg
Copyright-Jahr
2017
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-55350-3_11