Advertisement

A New Bound for the Existence of Differential Field Extensions

  • Richard GustavsonEmail author
  • Omar León Sánchez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

We prove a new upper bound for the existence of a differential field extension of a differential field \((K,\varDelta )\) that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in terms of lengths of certain antichain sequences of \({\text {I}\!\text {N}}^m\) equipped with the product order. This result has had several applications to effective methods in differential algebra such as the effective differential Nullstellensatz problem. Using a new approach involving Macaulay’s theorem on the Hilbert function, we produce an improved upper bound.

Keywords

Algebraic theory of differential equations Fields with several commuting derivations Differential field extensions 

References

  1. 1.
    Freitag, J., León Sánchez, O.: Effective uniform bounding in partial differential fields. Adv. Math. 288, 308–336 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gustavson, R., Kondratieva, M., Ovchinnikov, A.: New effective differential Nullstellensatz. Adv. Math. 290, 1138–1158 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lando, B.A.: Jacobi’s bound for the order of systems of first order differential equations. Trans. Am. Math. Soc. 152(1), 119–135 (1970)MathSciNetzbMATHGoogle Scholar
  4. 4.
    León Sánchez, O., Ovchinnikov, A.: On bounds for the effective differential Nullstellensatz. J. Algebra 449, 1–21 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26(2), 531–555 (1927)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Moreno Socías, G.: An ackermannian polynomial ideal. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, vol. 539, pp. 269–280. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  7. 7.
    Pierce, D.: Fields with several commuting derivations. J. Symb. Logic 79(01), 1–19 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsCUNY Graduate CenterNew YorkUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations