Not all elementary functions can be expressed with exp-minus-log
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Not all elementary functions can be expressed with exp-minus-log
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Re: Not all elementary functions can be expressed with exp-minus-log
#2The 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.
Re: Not all elementary functions can be expressed with exp-minus-log
#3The 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.
His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions.
I think it really comes down to what set of functions you are calling "elementary".
Re: Not all elementary functions can be expressed with exp-minus-log
#4The 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.
Yes, that blog post could have been much shorter….
Re: Not all elementary functions can be expressed with exp-minus-log
#5The 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.
Re: Not all elementary functions can be expressed with exp-minus-log
#6The 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 argument is that a universal basis would be capable of solving arbitrary polynomial roots. The rest is an argument that the group constructed by eml is solveable, and hence not all the standard elementary functions.
It wouldn't be a math discussion without people using at least two wildly different definitions.
Re: Not all elementary functions can be expressed with exp-minus-log
#7The 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.
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Re: Not all elementary functions can be expressed with exp-minus-log
#8I'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.
Re: Not all elementary functions can be expressed with exp-minus-log
#9The 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.
Can anyone provide a link that "Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this", or anything similarly grandiose?
I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...
Re: Not all elementary functions can be expressed with exp-minus-log
#10I'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.
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