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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Freitag, J., León Sánchez, O.: Effective uniform bounding in partial differential fields. Adv. Math. 288, 308–336 (2016)
Gustavson, R., Kondratieva, M., Ovchinnikov, A.: New effective differential Nullstellensatz. Adv. Math. 290, 1138–1158 (2016)
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)
León Sánchez, O., Ovchinnikov, A.: On bounds for the effective differential Nullstellensatz. J. Algebra 449, 1–21 (2016)
Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26(2), 531–555 (1927)
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)
Pierce, D.: Fields with several commuting derivations. J. Symb. Logic 79(01), 1–19 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Gustavson, R., Sánchez, O.L. (2016). A New Bound for the Existence of Differential Field Extensions. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-32859-1_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32858-4
Online ISBN: 978-3-319-32859-1
eBook Packages: Computer ScienceComputer Science (R0)