Not all elementary functions can be expressed with exp-minus-log
by mmastrac on 4/15/2026, 1:59:18 AM
https://www.stylewarning.com/posts/not-all-elementary/
Comments
by: 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.<p>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.<p>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: <a href="https://en.wikipedia.org/wiki/Schanuel%27s_conjecture" rel="nofollow">https://en.wikipedia.org/wiki/Schanuel%27s_conjecture</a><p>What is a closed-form number?: <a href="http://timothychow.net/closedform.pdf" rel="nofollow">http://timothychow.net/closedform.pdf</a> omega constant: <a href="https://en.wikipedia.org/wiki/Omega_constant" rel="nofollow">https://en.wikipedia.org/wiki/Omega_constant</a>
4/15/2026, 5:32:53 AM
by: 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.<p>> Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them.<p>If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.<p>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.
4/15/2026, 5:47:52 AM
by: 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.<p>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.
4/15/2026, 5:09:49 AM
by: derriz
When I first read the exp-minus-log paper, I found it extremely surprising - even shocking that such a function could exist.<p>But the fact that a single function can represent a large number of other functions isn't that surprising at all.<p>It's probably obvious to anyone (it wasn't initially to me), but given enough arguments I can represent any arbitrary set of n+1 functions (they don't even have to be functions on the reals - just as long as the domain has a multiplicative zero available) as a sort of "selector":<p>g(x_0, c_0, x_1, c_1, ... , x_n, c_n) = c_0 * f_0(x_0) + ... + c_n * f_n(x_n)<p>The trick is to minimize the number of arguments and complexity of the RHS - but that there's a trivial upper-bound (in terms of number of arguments).
4/15/2026, 6:08:24 AM
by: 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.<p>Can anyone please explain this further? It seems like he’s moving the goalposts.
4/15/2026, 4:57:58 AM
by: 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.<p>AFAIU the original paper is a result in the field of symbolic regression. What definition of elementary function do they use?
4/15/2026, 5:37:31 AM
by: bawolff
> Elementary functions typically include arbitrary polynomial roots<p>Admittedly this may be above my math level, but this just seems like a bad definition of elementary functions, given the context.
4/15/2026, 5:30:46 AM
by: avmich
I'd really like more details on the terminology used.<p>Also I'd be glad to see a specific example of a function, considered elementary, which is not representable by EML.<p>It could be hard, and in any case, thanks for the article. I wish it would be more accessible to me.
4/15/2026, 5:11:55 AM
by: renewiltord
If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century.
4/15/2026, 5:40:51 AM