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

> 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.

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.

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

That's a kind of weak criticism. What functions are considered elementary was always going to be arbitrary, picking the set you can generate from exp, log, and some complex algebra is not the worst choice.

If nothing else you could solve simple differential equations with them. And it gives you the 'power' function.

The very fact that the set of functions is largely arbitrary is a much bigger issue. Or at least it limits the use of the fact that you can represent those functions.

Edit: I feel the need to add that just because it is a weak critique doesn't mean the argument itself is not interesting.

When I first read the exp-minus-log paper, I found it extremely surprising - even shocking that such a function could exist.

But the fact that a single function can represent a large number of other functions isn't that surprising at all.

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":

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)

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).

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.

It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators.

Interestingly, the abs (absolute value) function is non-elementary. I wonder if exp-minus-log can represent it.

> 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.

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?

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.

tangenially it would be interesting to analyze what infinite structures can be represented with infinite EML trees

On a tangent: I've tried to connect Euclid's Elements with quantifier elimination theorems. It looks like most of the geometry follows from QE of real-closed fields. Some of the number theory relates to Presburger arithmetic. Some other number theory, including the irrationality of sqrt(2), is down to Skolem. The Pythagorean triples relate to extending Skolem to the Gaussian integers. I suspect some of the "embryonic" integral calculus could be related to holonomic functions, which seem like they admit a form of QE.

Don't have anything for the perfect numbers though.

Where would EML expressions sit in this fascinating table?

https://en.wikipedia.org/wiki/Template:Mathematical_expressi...

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