Not all elementary functions can be expressed with exp-minus-log

lotaezenwa

The author essentially says that the quintic has no closed form solution which is true regardless of the exp-minus-log function. The purpose of this blog post is lost on me. Can anyone please explain this further? It seems like he’s moving the goalposts.

saithound

The original article explicitly acknowledged this limitation, that while 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, i.e., adjoining roots of polynomial equations," the author works with the less general definition. Neither the present article, nor the original one has much mathematical originality, though: Odrzywolek's result is immediately obvious, while this blog post is a rehash of Arnold's proof of the unsolvability of the quintic.

avmich

I'd really like more details on the terminology used. Also I'd be glad to see a specific example of a function, considered elementary, which is not representable by EML. It could be hard, and in any case, thanks for the article. I wish it would be more accessible to me.

bawolff

> Elementary functions typically include arbitrary polynomial roots Admittedly this may be above my math level, but this just seems like a bad definition of elementary functions, given the context.

SabrinaJewson

Related is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this. In a similar vein to this post, the paper points out that general polynomials do not have solutions in E, so of course exp-minus-log is similarly incomplete. What is intriguing is that we don’t even know whether many simple equations like exp(-x) = x (i.e. the [omega constant]) have solutions in E. We of course suspect they don’t, but this conjecture is not proven: https://en.wikipedia.org/wiki/Schanuel%27s_conjecture What is a closed-form number?: http://timothychow.net/closedform.pdf omega constant: https://en.wikipedia.org/wiki/Omega_constant

zarzavat

This is a bit like invalidating a result based on 0^0 := 1 because you work in a field of mathematics where 0^0 is an indeterminate form. Not very interesting. AFAIU the original paper is a result in the field of symbolic regression. What definition of elementary function do they use?

renewiltord

If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century.

rnhmjoj

> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold. > Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them. If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I've actually just learnt that some consider roots of arbitrary polynomials being part of the elementary functions before, but I'm a physicist and only ever took some undergraduate mathematics classes. Nonetheless, calling these elementary feels a bit of stretch considering that the word literally means basic stuff, something that a beginner will learn first.