Not all elementary functions can be expressed with exp-minus-log
9 points by Corbin
9 points by Corbin
The paper that this is responding to anticipates the point of this blog post on page 5:
In the classical differential-algebraic setting, one often works with a broader notion of elementary function, defined relative to a chosen field of constants and allowing algebraic adjunctions [7], i.e., adjoining roots of polynomial equations (ef. Root in Wolfram Language [33]). That level of generality is not needed here. The present paper takes the ordinary scientific-calculator point of view: start from a concrete list of familiar constants, functions and operations, and ask how far they can be reduced without losing practical functionality. The precise starting list is given later in Table 1.
The blog post that you are responding to anticipates this comment:
My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage. Odrzywołek recognizes this, saying little more than “[t]hat generality is not needed here” and that his work takes “the ordinary scientific-calculator point of view”. He does not offer further commentary.
What is this more general setting, and does his claim still hold? In modern pure mathematics, dating back to the 19th century, the definition of “elementary function” has been well established. We’ll get to a definition shortly, but to cut to the chase, the titular result does not hold in this setting. As such, in layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates.
The rough TL;DR is this: Elementary functions typically include arbitrary polynomial root functions, and EML terms cannot express them.
This whole thing is a tedious quibble about the meaning of “elementary”, and the lemmas about monodromy groups are irrelevant. The original paper is clear that the author meant it as high school maths, as embodied by the buttons on a calculator. For instance the fourth paragraph of the introduction says,
Elementary functions, for many students epitomized by the dreaded sine and cosine
No one would write that if their target audience considers the roots of quintics to be elementary.
Perhaps it was a mistake to delay the jargon clarification to page 5. Mathematicians have weird tropes about titles, where “fundamentals” or “elementary” doesn’t mean what you learn first, they mean what you get when you deconstruct advanced knowledge to its constituent parts.
I would like this article much better if it suggested a word to use instead of “elementary”. How do mathematicians name the vocabulary of functions in math.h or on my calculator buttons?