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Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions

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Mathematical Aspects of Computer and Information Sciences (MACIS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11989))

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Abstract

Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \(\eta \) of its difference-differential generators and prove that for any intermediate difference-differential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial \(\chi _{K, F,\eta }(t)\). This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of Kähler differentials \(\varOmega _{L^{*}|K}\) where \(L^{*}\) is the inversive closure of L. We prove some properties of polynomials \(\chi _{K, F,\eta }(t)\) and use them for the study of the Krull-type dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations.

Supported by the NSF grant CCF-1714425.

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Authors and Affiliations

  1. The Catholic University of America, Washington, DC, 20064, USA

    Alexander Levin

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  1. Alexander Levin
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Correspondence to Alexander Levin .

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Editors and Affiliations

  1. AIT Austrian Institute of Technology, Vienna, Austria

    Daniel Slamanig

  2. IMJ-PRG, Sorbonne University, Paris, France

    Elias Tsigaridas

  3. Institute of Information Technologies, Gebze Technical University, Gebze, Turkey

    Zafeirakis Zafeirakopoulos

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Levin, A. (2020). Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions. In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science(), vol 11989. Springer, Cham. https://doi.org/10.1007/978-3-030-43120-4_7

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  • DOI: https://doi.org/10.1007/978-3-030-43120-4_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43119-8

  • Online ISBN: 978-3-030-43120-4

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