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Leibniz complexity of Nash functions on differentiations
- Publication Year :
- 2015
-
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 K{\" a}hler 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.<br />Comment: 19 pages, and we will submit this article for some mathematics journal
- Subjects :
- Mathematics - Algebraic Geometry
14P20 (Primary), 58A07 (Secondary), 32C07
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1509.08261
- Document Type :
- Working Paper