Leibniz complexity of Nash functions on differentiations

TL;DR

This paper introduces the Leibniz complexity for Nash functions, measuring the minimal number of Leibniz rule applications needed to algebraically compute derivatives, and shows that non-Nash functions cannot be derived algebraically.

Contribution

It defines Leibniz complexity for Nash functions and establishes bounds and properties, highlighting the algebraic limitations for derivatives of non-Nash functions.

Findings

Leibniz complexity provides a measure of derivative computation complexity.

Non-Nash analytic functions cannot have their derivatives derived algebraically.

Upper bounds relate Leibniz complexity to binary expansions and other complexity measures.

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.