Student solves ancient math problem

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Discussion Overview

The discussion revolves around a Dutch student's claim of having solved an ancient mathematical problem related to finding the zero-points of polynomials of any degree. Participants explore the implications of this claim in the context of historical contributions to polynomial equations, particularly focusing on the work of mathematicians like Galois and Abel.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express skepticism about the validity of the student's formula, questioning how it aligns with established mathematical theories, particularly Galois' work.
  • Others highlight the historical context of polynomial equations, noting that no general formula exists for polynomials of degree greater than four, as established by Abel's theorem.
  • A participant mentions the difficulty of understanding the student's paper and suggests it lacks clarity and rigor.
  • Some contributors discuss numerical methods used by graphing calculators to find polynomial roots, raising questions about the relationship between these methods and the existence of a general formula.
  • There are references to Galois' contributions to finite fields and the implications of his work on modern mathematics.
  • A few participants share personal experiences with polynomial equations, indicating a range of understanding and engagement with the topic.
  • One participant proposes that while a general formula may not exist, algorithms could potentially work for a dense subset of polynomials.
  • Another participant mentions their own theorem related to cubic polynomials, suggesting a need for clarity in mathematical terminology.

Areas of Agreement / Disagreement

Participants generally do not reach a consensus on the validity of the student's claim. There are multiple competing views regarding the implications of the student's work, the historical context of polynomial equations, and the clarity of the presented paper.

Contextual Notes

Limitations in understanding the student's paper are noted, with some participants indicating that it may not adequately prove its claims. The discussion also reflects varying levels of familiarity with polynomial theory and historical mathematical concepts.

Monique
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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 :biggrin:
 
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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...
 
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)??
 
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
 
I could not help reading it :wink:
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.
 
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?
 
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.
 
ChanDdoi said:
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
 
  • #11
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
 
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  • #12
I was referring to incidence-geometry in my last post

mon cher ami humanino

regards
marlon
 
  • #13
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:
 
  • #14
humanino said:
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 :blushing: et l'italien :cool: .
Duncque puoi scegliere tu...

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


regards
marlon
 
  • #15
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:
 
  • #16
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.
 
  • #17
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.
 
  • #18
I was sure they would not even bother trying to search where it goes wrong, because it is too obviously wrong.

Hurkyl said:
... 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.
 
  • #19
jcsd said:
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
 
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  • #20
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.
 
  • #21
Loren Booda said:
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.
 
  • #22
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.
 
  • #23
I contacted the author, let's see what he thinks ;)
 
  • #24
thanks, shmoe, I will try making those corrections soon.
 
  • #25
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?
 
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  • #26
so really it's nothing to raise an eyebrow about
 
  • #27
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.
 
  • #28
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.
 
  • #29
Ethereal said:
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 :smile:.

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
 
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  • #30
Okay, thanks for the links!