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July, 2019 Leibniz complexity of Nash functions on differentiations
Goo ISHIKAWA, Tatsuya YAMASHITA
J. Math. Soc. Japan 71(3): 709-726 (July, 2019). DOI: 10.2969/jmsj/76877687

Abstract

The derivatives of Nash functions are Nash functions which are derived algebraically from their minimal polynomial equations. In this paper we show that, for any non-Nash analytic function, it is impossible to derive its derivatives algebraically, i.e., by using linearity and Leibniz rule finite times. In fact we prove the impossibility of such kind of algebraic computations, algebraically by using Kähler differentials. Then the notion of Leibniz complexity of a Nash function is introduced in this paper, as a computational complexity on its derivative, by the minimal number of usages of Leibniz rules to compute the total differential algebraically. We provide general observations and upper estimates on Leibniz complexity of Nash functions, related to the binary expansions, the addition chain complexity, the non-scalar complexity and the complexity of Nash functions in the sense of Ramanakoraisina.

Funding Statement

The first author was supported by JSPS KAKENHI Grant No.JP15H03615 and No.JP15K13431.

Citation

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Goo ISHIKAWA. Tatsuya YAMASHITA. "Leibniz complexity of Nash functions on differentiations." J. Math. Soc. Japan 71 (3) 709 - 726, July, 2019. https://doi.org/10.2969/jmsj/76877687

Information

Received: 14 December 2016; Revised: 18 January 2018; Published: July, 2019
First available in Project Euclid: 18 March 2019

zbMATH: 07121550
MathSciNet: MR3984239
Digital Object Identifier: 10.2969/jmsj/76877687

Subjects:
Primary: 14P20
Secondary: 32C07 , 58A07

Keywords: Complexity , extension theorem of Efroymson , Kähler differentials

Rights: Copyright © 2019 Mathematical Society of Japan

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Vol.71 • No. 3 • July, 2019
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