Student solves ancient math problem

[Monique]

A dutch student from Eindhoven, G. Uytdewilligen, solved an ancient mathematical problem: after two years of struggling he came up with a formula that describes the zero-points (where the function crosses the y-axes) of polynomals of ANY degree.

Only during the Renaissance did Gerolamo Gardano (1501-1576) solve the equation for 3rd degree polynomals. Ferrari (1522-1565) solved the 4th degree equation, Galois (1811-1832) classified the 'unsolvable' 5th degree polymals with his grouptheory.

Noone was able to come up with an answer for higher-degree polynomals, but this student just wrote a formula that solves it for ANY polynomal

 

[humanino]

This is a great info Monique ! Thank you very much.

However, what happened to Galois' theory ? I doubt...
See : pdf on arXiv
I am going to read it when have time. Please somebody comment...

 

[jcsd]

Here's the PDF, it's only a little over two pages long (they might also have it in French):

http://hal.ccsd.cnrs.fr/docs/00/00/32/74/PDF/The-roots-of-any-polynomial-equation.pdf

Colour me sceptical.

 

[ChanDdoi]

considering me ignorant if you like, but i was just wondering ...if there is no formula to find the roots of high degree polynomial ...then how can the graphing calculators do it (i mean like how was it programed to find the roots)??

 

[A_I_]

that's great..
i tried once.. i focused for a while..
i tried to solve the high-degrees polynomials functions..
i have some notions..
but i don't think they're enough

 

[humanino]

I could not help reading it
Alas, I was not able to understand it yet. I think such an important would require a higher quality paper. I would advise this student to write a better review.
I am eager to see mathematicians response.

 

[jcsd]

Yep, I'm having real problem trying to understand exactly the steps he took, as the pape'rs not very well written. Hoepfully Matt Grime can take a look at it, but the more I read of it the more skeptical I become.

Out of interest where did you get the story from Monique?

 

[humanino]

Could one become more skeptical after reading ? Galois is among the greatest mathematicians of all time.

Just a thought : maybe there is no method for general polynomials, but one can prove that an algorithm works for a dense subset in the polynomial space. That would be useful.

 

[marlon]

considering me ignorant if you like, but i was just wondering ...if there is no formula to find the roots of high degree polynomial ...then how can the graphing calculators do it (i mean like how was it programed to find the roots)??

I think the answer to your question is numerical analysis...

regards
marlon

 

[cronxeh]

i donno what the hell he tried to solve there.. and also it looks just like http://mathworld.wolfram.com/VietasFormulas.html

 

[marlon]

Humanino,

This Galois, isn't he the guy from the finite fields in Algebra. Stuff like GF(2) where you need to calculate every number modulo 2. So only 0 and 1 are valid numbers.

That's very nice because you get crazy results in such algebraic structure like the Moulton-plane or the Hilbert-plane. When you work with kar2 (modulo 2) the line y = x is PARALLEL to y = -x since 1 and -1 are equivalent
(-1 = 1 mod2).

nice, isn't it ?

regards
marlon

 

[marlon]

I was referring to incidence-geometry in my last post

mon cher ami humanino

regards
marlon

 

[humanino]

Hey Marlon !
I am not too sure about everything you wrote. But yes, he is the guy from the finite fields in Algebra.
Galois

Some people (including me) think that if he did not die so young, math would be totally different today. He was an extraordinary genius, and such a compliment is even unfair to him... His work is really difficult to grasp, I tried and did not fully succeded. His life is fascinating.

Mon cher Marlon, j'aimerais pouvoir vous faire l'honneur de vous repondre dans votre langue :shy:

 

[marlon]

Hey Marlon !
I am not too sure about everything you wrote. But yes, he is the guy from the finite fields in Algebra.
Galois

Some people (including me) think that if he did not die so young, math would be totally different today. He was an extraordinary genius, and such a compliment is even unfair to him... His work is really difficult to grasp, I tried and did not fully succeded. His life is fascinating.

Mon cher Marlon, j'aimerais pouvoir vous faire l'honneur de vous repondre dans votre langue :shy:

Je parle néerlandais , mon cher humanino, mais aussi le français et l'italien .
Duncque puoi scegliere tu...

Trust me, it is no joke what I wrote about these modulo-things. We studied it in college...


regards
marlon

 

[humanino]

Yes I trust you. We probably don't have the same notations/semantics that is all. In France, fields are called "corps" (bodies) for instance. This avoid possible confusion between math fields and phys. fields. Of course, such a confusion is very unlikely to happen to somebody knowing the context.

EDIT Sprechen Sie nicht Deutcsh ? ... oder ähnlich Sache :shy:

 

[Zurtex]

Has this been disproved yet?

Sorry to be so sceptical but the paper is awful and doesn't look like ground breaking mathematics to me. In fact reading that over again it doesn't even seem to prove anything.

 

[Hurkyl]

Only during the Renaissance did Gerolamo Gardano (1501-1576) solve the equation for 3rd degree polynomals. Ferrari (1522-1565) solved the 4th degree equation, Galois (1811-1832) classified the 'unsolvable' 5th degree polymals with his grouptheory.

Noone was able to come up with an answer for higher-degree polynomals, but this student just wrote a formula that solves it for ANY polynomal

To put this into perspective:

Abel's theorem is that for any n > 4, there is no formula for the roots of a general n-th degree polynomial that uses only the coefficients of the polynomial, arbitrary integers, +, -, *, /, and k-th roots.

Furthermore, one can find explicit polynomials (I think x^5 + 4x + 2 is one) whose solutions cannot be expressed in terms of integers, +, -, *, /, and k-th roots.

 

[humanino]

I was sure they would not even bother trying to search where it goes wrong, because it is too obviously wrong.

... one can find explicit polynomials (I think x^5 + 4x + 2 is one) whose solutions cannot be expressed in terms of integers, +, -, *, /, and k-th roots.

right, we should have thought about that earlier.

 

[Monique]

Out of interest where did you get the story from Monique?

A friend of mine pointed me to the article http://www.nu.nl/news.jsp?n=404623&c=80

 

[Loren Booda]

Look for my "Booda Theorem" on my website, below. It involves a simply stated, nontrivial, and original proof involving cubic polynomials - but nothing to change the world.

 

[shmoe]

Look for my "Booda Theorem" on my website, below. It involves a simply stated, nontrivial, and original proof involving cubic polynomials - but nothing to change the world.

Hi, you seem to have left out the important word "local" in the statement of your result, a cubic polynomial has no minimum or maximum. Every author I've seen uses the convention that without the qualifier of "local" you are referring to absolute extrema on your entire domain when you say "maximum" or "minimum". You also don't need the "and"-if it has a local min then it has a local max and vice versa (this is true for any odd degree polynomial). I'd also object to your use of the word "nontrivial" as the proof is straightforward to anyone 2 months into their first calculus course once you see the statement of the result. Not to take anything away from you- a 16 year old who learns calc is impressive by any standards imo, and it was an interesting idea to examine the relation between the critical points and inflection point (anything for higher degree polynomials?).



Back to the topic at hand...I had a look at that paper and it's not the easiest thing in the world to follow what he's attempting to do. It appears he gets the roots (or maybe just one root?) of the polynomial as a power series whose coefficients can be solved for and depend on the original polynomial (he calls it a power series, but I'm not sure if it's intended to have a finite number of terms his eq (5)). I'm not willing to spend much time deciphering this paper, but if the end result is to get the roots as a power series then this in no way contradicts Abel/Galois result on insolubility. See Hurkyl's post for more details on what operations Abel/Galois allow and I'd like to point out that they were also limited a finite number of them (i.e. infinite series is not allowed).

If his end result is claiming to contradict Abel/Galois, then he's wrong. I'd be willing to cut off my left hand if he does successfully contradict the old result on insolubility.

 

[matt grime]

He isn't contradiciting galois theory. He requires summing an infinite series (he talks about the convergence issues), and it produces another non-analytic way of finding roots of polynomials. It looks a lot less efficient than the other ones we know and love, and indeed it only says it finds a set of numbers, some of which are the roots, so even after running it you need to check back.

 

[Monique]

I contacted the author, let's see what he thinks ;)

 

[Loren Booda]

thanks, shmoe, I will try making those corrections soon.

 

[modmans2ndcoming]

he uses infinite series to approximate the roots. sure, you can get close enough, but please it is not a general solution for polynomials of the nth degree.

edit-

woops look like Matt got to it before I posted. it is sort of funny though how all these press releases make it seem like the kid is a fields candidate.

so, if I use a taylor series in order to integrate ln(x), does that mean that I can get a fancy press release?

 

[gazzo]

so really it's nothing to raise an eyebrow about

 

[matt grime]

No, not in the manner that has been implied by many writers. Galois theory is correct. It is easy to learn and one of the nicest parts of pure mathematics there is, and it is not hard to show that in general polynomials do not have roots that are expressible algebraically in terms of the coefficients. Anyone who claims otherwise ought to take a step back.

It may be that htis student has found a *good* numerical approach (I have yet to see the student making any claims about contradicting galois theory; surely he'd've put that in the abstract if he actually thought that was what he was doing). As it only works for finding zeroes of polynomials there should be something useful about it, to make it different from the other methods of fidning zeroes of more general function, other wiase I don't even see why he'd have written it down.

 

[Ethereal]

A quick question:
Does anyone know of any good web sites which contain a large collection of mathematical proofs? It's a little frustrating having to search the Web with Google only to find nothing.

 

[da_willem]

A quick question:
Does anyone know of any good web sites which contain a large collection of mathematical proofs? It's a little frustrating having to search the Web with Google only to find nothing.

Try Google .

On this site you will find a large collection of mathematical proofs: http://list-of-mathematical-proofs.wikiverse.org/ (there are a lot of sites containing the same list, so sorry if this is a list you have already found). A little less advanced mathematics: http://www.cut-the-knot.org/proofs/index.shtml .

Some mirror sites of the first site mentioned:
http://encyclopedia.thefreedictionary.com/list of mathematical proofs
http://www.campusprogram.com/reference/en/wikipedia/l/li/list_of_mathematical_proofs.html
http://www.book-spot.co.uk/index.php/List_of_mathematical_proofs

 

[Ethereal]

Okay, thanks for the links!

 

[Orion1]

Galois Giant...

Evariste Galois (1811-1832), a French mathematician, proved that it is not possible to solve a 'general' fifth (or higher) degree polynomial equation by radicals. Of course, we can solve particular fifth-degree equations such as x^5 - 1 = 0, but Abel and Galois were able to show that no general 'radical' solution exists.

Galois's Theorem:
An algebraic equation is algebraically solvable if its group is solvable. In order that an irreducible equation of prime degree be solvable by radicals, it is necessary and sufficient that all its roots be rational functions of two roots.

Evariste Galois was 21 years old when he died.

Abel's Impossibility Theorem:
In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (Wells 1986, p. 59).

Abel-Ruffini theorem:
...states that there is no general solution in radicals to polynomial equations of degree five or higher.

In the modern analysis, the reason that second, third and fourth degree polynomial equations can always be solved by radicals while higher degree equations cannot is nothing but the algebraic fact that the symmetric groups S2, S3 and S4 are solvable groups, while Sn is not solvable for n=>5.

Reference:
Calculus - (Larsen, Hosteller, Edwards) - Fourth Edition - pg.238
http://mathworld.wolfram.com/GaloissTheorem.html
http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html
http://en.wikipedia.org/wiki/Abel-Ruffini_theorem
http://c2.com/cgi/wiki?FermatsLastTheorem
http://proofs-of-fermat-s-little-theorem.wikiverse.org/
http://en.wikipedia.org/wiki/Symmetric_group