Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

stylewarning.com

Post reply on HN
Print view View original

41–50 of 125 posts

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

#41

Post by cyberax »

derriz wrote: post #29

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 t…

Why would it be surprising?

And if you want something truly surprising, Riemann's zeta function can approximate any holomorphic function arbitrarily well on the critical strip. So technically you need only _one_ argument.

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

#43

Post by saithound »

reikonomusha wrote: post #35
saithound wrote: post #5

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 ha…

Arnold (as reported by Goldmakher [1]) does prove the unsolvability of the quintic in finite terms of arithmetic and single-valued continuous functions (which does not include the complex logarithm). TFA's result is stronger, which is something about the solvability of the monodromy groups of all EML-derived functions. So it doesn't seem to be a "rehash", even if their specific counterexample could have been achieved…

Arnold's proof can be used to show that certain classes of functions are insufficient to express a quintic formula.

These classes can always safely include all single-valued continuous functions (you cannot even write the _quadratic_ formula in terms of arithmetic and single-valued continuous functions!), but also plenty of non-single-valued functions (e.g. the +-sqrt function which appears in the well-known quadratic formula).

Applying Arnold's proof to the class given by arithmetic and all complex nth root functions (also multivalued) gives the usual Abel-Ruffini theorem. But Arnold's proof applies to the class "all elm-expressible functions" without modification.

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

#44

Post by js8 »

bawolff wrote: post #13

> 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.

I would agree, it makes them anything but elementary. I am honestly not even sure if there is a finite constructible basis of the functions that can express any solution of single-variable integer polynomials.

And for multivariate polynomials, the roots are uncomputable due to MRDP theorem.

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

#45

Post by cyberax »

reikonomusha wrote: post #15
lotaezenwa wrote: post #2

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.

"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical. The post's argument is di…

Bring radicals are just cheating.

You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.

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

#46

Post by chii »

rnhmjoj wrote: post #24

> 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 lea…

jargon are words being used that don't carry the typical laymen definition, but a specific one from the domain of said jargon.

If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate.

But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?

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

#47

Post by reikonomusha »

cyberax wrote: post #45
reikonomusha wrote: post #15

Earlier quoted context omitted.

"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical. The post's argument is di…

Bring radicals are just cheating. You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.

This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known.

Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved.

Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.

A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc.

The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was:

    Do integrals of elementary functions give us elementary functions?
The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary.

Risch gave us an algorithm to compute the answer, when it exists in elementary form.

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

#48

Post by throwaway81523 »

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.

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

#49

Post by ogogmad »

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.

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

#50

Post by mcmoor »

rnhmjoj wrote: post #24

> 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 lea…

I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?
Post reply on HN