[markgall]

I only skimmed the article, but I think the idea is to use some variation on:

f(a,b,c,d,e) = the largest real solution x of the quintic equation x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0

There's not a simple formula for this function (which is the basic point), but certainly it is a function: you feed it five real numbers as input, and it spits out one number as output. The proof that you can't generate this function using the single one given looks like some fairly routine Galois theory.

Whether this function is "considered elementary" depends on who you ask. Most people would not say this is elementary, but the author would like to redefine the term to include it, which would make the theorem not true anymore.

Why any of this would shake the foundations of computer engineering I do not know.

[avmich]

I've thought something like that, but I'm interested more in details of the argument.

As for why this could be important... we sometimes find new ways of solving old problems, when we formulate them in a different language. I remember how i was surprised to learn how representation of numbers as a tuple (ordered list of numbers), where each element is the remainder for mutually prime dividers - as many dividers as there are elements in the tuple - reduces the size of tables of division operation, and so the hardware which does the operation using thise tables may use significantly less memory. Here we might have some other interesting advantages.

[Aardwolf]

But can you even express this function with the elementary operator symbols, exp, log, power and trig functions? It seems to me like no, you can't express "largest real solution" with those (and what's the intended result for complex inputs?)

At least eml can express the quintic itself, just like the above mentioned operators can

[tliltocatl]

Author and EML are using different definitions of elementary functions, EML's definition being the school textbooks' one (polynomials, sin, exp, log, arcsin, arctan, closed under multiplication, division and composition). The author's definition I've never met before, it apparently includes some multi-valued functions, which are quite unusual.

[reikonomusha]

Wikipedia says:

> More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary. [...] This list of elementary functions was originally set forth by Joseph Liouville in 1833.

which seems to be what the blog post references.

[teo_zero]

I feel that saying that EML can't generate all the elementary functions because it can't express the solution of the quintic is like saying that NAND gates can't be the basis of modern computing because they can't be used to solve Turing's halting problem.

[reikonomusha]

As is usual with these kinds of "structure theorems" (as they're often called), we need to precisely define what set of things we seek to express.

A function which solves a quintic is reasonably ordinary. We can readily compute it to arbitrary precision using any number of methods, just as we can do with square roots or cosines. Not just the quintic, but any polynomial with rational coefficients can be solved. But the solutions can't be expressed with a finite number of draws from a small repertoire of functions like {+, -, *, /}.

So the question is, does admitting a new function into our "repertoire" allow us to express new things? That's what a structure theorem might tell us.

The blog post is exploring this question: Does a repertoire of just the EML function, which has been shown by the original author to be able to express a great variety of functions (like + or cosine or ...) also allow us to express polynomial roots?

[zeroonetwothree]

That’s a poor analogy because all polynomials can be solved to arbitrary precision with efficient algorithms.