# What is the state of the art in solving polynomial equations?

• A
Hi folks,

I know the work from Galois showing it is not possible to solve certain equations using only a certain type of numbers. But that was more than 100 years ago, I suppose lots of progress has been made on this topic.

So my question is, What has been discovered to solve polynomial equations? I don't mean algorithmic techniques, I mean theoretical calculations and exact solutions.

He showed some equations are not solvable using specific numbers, Has it been solved using another kind of numbers or something like that?

If I give you a degree 500 polynomial, Do you have the mathematical techniques to solve the equation and find the 500 solutions? I don't mean numerical calculations.

## Answers and Replies

fresh_42
Mentor
Hi folks,

I know the work from Galois showing it is not possible to solve certain equations using only a certain type of numbers. But that was more than 100 years ago, I suppose lots of progress has been made on this topic.

So my question is, What has been discovered to solve polynomial equations? I don't mean algorithmic techniques, I mean theoretical calculations and exact solutions.

He showed some equations are not solvable using specific numbers, Has it been solved using another kind of numbers or something like that?

If I give you a degree 500 polynomial, Do you have the mathematical techniques to solve the equation and find the 500 solutions? I don't mean numerical calculations.
Old or not, Galois' theory says there is no way to express the roots of arbitrary polynomials over the integers of degree higher than four by radicals, i.e. root expressions. There have been proofs for the case of degree five other than Galois' but his theory can be considered as the most elegant and shortest.

It is not a matter of time. It is a matter of fact.

This does not mean that one cannot find roots for special polynomials of any degree. E.g. multiply ##(x-1)\cdot (x-2) \cdot \dots \cdot (x-500)## and I will tell you the roots. Galois' theory also supplies a criterion which polynomials are solvable. There is nothing to be found except better numerical or algorithmic solutions.

• PsychonautQQ and jonjacson
But there must have been progress right? What are the unanswered questions in this topic?

I have read something about grobner basis but I don't get the point. Apparently they are just another representation for the same polynomial that looks more simple.

Is this correct?

What is the people working on this topic investigating?

fresh_42
Mentor
It seems that there is a general solution for how polynomials of fifth degree have to look like to be solvable (Young, Runge 1885). You might read about it here or in general here on Wikipedia. There are also a lot of links and sources there to extend studies.
I'm not quite sure how Gröbner basis help but they certainly belong into the same algebraic context. As far as I know they are mainly used for algorithmic (computational) approaches and in the field of algebraic varieties which is far more general.

Since our computational abilities are so far developed and the general underlying question has been solved, I doubt there is much effort to solve rational equations in just one variable of higher (than five) degrees, resp. describe the coefficients to determine whether they are solvable or not. There might be some results for degrees six or seven.

• PsychonautQQ
hilbert2
Science Advisor
Gold Member
It probably wouldn't produce any interesting new information, if we tried to define new special functions ad hoc only to be able to solve polynomial equations of higher order. I personally find questions related to the stability of polynomials (how much the locations of the roots change if the coefficients of the polynomial undergo a small change) a lot more interesting.

It seems that there is a general solution for how polynomials of fifth degree have to look like to be solvable (Young, Runge 1885). You might read about it here or in general here on Wikipedia. There are also a lot of links and sources there to extend studies.
I'm not quite sure how Gröbner basis help but they certainly belong into the same algebraic context. As far as I know they are mainly used for algorithmic (computational) approaches and in the field of algebraic varieties which is far more general.

Since our computational abilities are so far developed and the general underlying question has been solved, I doubt there is much effort to solve rational equations in just one variable of higher (than five) degrees, resp. describe the coefficients to determine whether they are solvable or not. There might be some results for degrees six or seven.

But the underlying question was not resolved.

I mean, what Galois showed is how you can't solve the equation. But he didn't show how to solve the equation. I cannot solve the equations using a paper and a pencil.

Maybe the question I am asking in fact is, Is it possible to generate the real numbers using the rational+irrational+ or any combination of parts of them?

It probably wouldn't produce any interesting new information, if we tried to define new special functions ad hoc only to be able to solve polynomial equations of higher order. I personally find questions related to the stability of polynomials (how much the locations of the roots change if the coefficients of the polynomial undergo a small change) a lot more interesting.

Yes, I am talking about that. Defining a new special function or number, or combination of small parts of numbers to solve it.

Why do you think it is interesting?

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SteamKing
Staff Emeritus
Science Advisor
Homework Helper
I think you are missing the point of how polynomial equations of degrees one thru four were solved and why this method of solution could not be extended to polynomials of degree five and higher. Galois theory showed how the regular method of solution worked for less complex polynomials and why it could not be utilized in the general case for equations of degree five and above.

For solving low-order equations, there was always some formula (like the quadratic) or procedure which involved manipulating the coefficients of the polynomial, and the solution came down to extracting a square or higher root of the manipulated set of coefficients. The coefficients of the equation are thus related to the roots of the equation in some manner and to each other as well.

https://en.wikipedia.org/wiki/Galois_theory

Now, Galois theory answers the narrow question posed thus:

Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?

Galois theory says that a general formula or algorithm is not constructible, using the limitations described above, for all polynomials of degree five and above. Obviously, there are some higher order equations which are solvable using clever substitutions, but these can be done only on a case-by-case basis.

It is interesting that Galois theory also answers some other unsolved questions of greater antiquity, such as why an angle cannot be trisected using a compass and straightedge alone.

The difficulty in understanding Galois theory is that you need to know more than just solving polynomials, you need to understand how the groups of solutions are related, and this requires a much higher level of abstract thought.

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• PsychonautQQ and fresh_42
fresh_42
Mentor
But he [Galois] didn't show how to solve the equation. I cannot solve the equations using a paper and a pencil.
You are right, Galois theory doesn't tell us how. Nevertheless it tells us whether at all! And it supplies criteria for the search. From a mathematical point of view this is a finite answer. The rest is simply a matter of more or less good algorithms. You might try Wolfram for some tests.

On the other hand the classification of solvable groups is a vast field and we still don't really know how to attack it. Maybe there is no suitable way because of the great variety of solvable groups. But even there exist many, many tools and theorems in which partially answers are given.

The variation of polynomial zeros in dependency on their coefficients is a totally different approach and mathematical field which does not need to know exact expressions of the roots.

• PsychonautQQ
Ok, I will have a look again at the theory and try to understand better it.

You are right, Galois theory doesn't tell us how. Nevertheless it tells us whether at all! And it supplies criteria for the search. From a mathematical point of view this is a finite answer. The rest is simply a matter of more or less good algorithms. You might try Wolfram for some tests.

On the other hand the classification of solvable groups is a vast field and we still don't really know how to attack it. Maybe there is no suitable way because of the great variety of solvable groups. But even there exist many, many tools and theorems in which partially answers are given.

The variation of polynomial zeros in dependency on their coefficients is a totally different approach and mathematical field which does not need to know exact expressions of the roots.

I will search information about that approach.

Thanks for you answers and I apologize if I am bothering someone with my ignorance-based questions. :)

You are right, Galois theory doesn't tell us how.

Actually, it does. Using Galois theory, you can find out whether a polynomial is solvable and if it is, it tells you exactly how to find the roots. It's just that the method is very cumbersome and leads to very long representations of roots. The method is implemented in GAP for example: http://www.gap-system.org/Manuals/pkg/radiroot/doc/manual.pdf

• PsychonautQQ, suremarc, fresh_42 and 1 other person
I think what jonjacson is trying to ask is: if radical expressions are insufficient to write exact solutions of quintic and higher equations, what mathematical tools would be sufficient?

For example, one can write exact solutions of a general quintic polynomial using Jacobi theta functions or generalized hypergeometric functions.

What tools are needed to go further?

• PsychonautQQ and jonjacson
I think what jonjacson is trying to ask is: if radical expressions are insufficient to write exact solutions of quintic and higher equations, what mathematical tools would be sufficient?

For example, one can write exact solutions of a general quintic polynomial using Jacobi theta functions or generalized hypergeometric functions.

What tools are needed to go further?
That is exactly what I haven't been able to explain.!!!!!!!!!!!!!!!

I haven't heard about Jacobi or hypergeometric, good info!

• PsychonautQQ
Galois theory says that a general formula or algorithm is not constructible, using the limitations described above, for all polynomials of degree five and above. Obviously, there are some higher order equations which are solvable using clever substitutions, but these can be done only on a case-by-case basis.

It is interesting that Galois theory also answers some other unsolved questions of greater antiquity, such as why an angle cannot be trisected using a compass and straightedge alone.

The difficulty in understanding Galois theory is that you need to know more than just solving polynomials, you need to understand how the groups of solutions are related, and this requires a much higher level of abstract thought.

First of all, thank you for your whole post, i found it very insightful. Secondly, when you say that the difficulty in understanding Galois theory is that you need to know more than just solving polynomials, do you mean to say that understanding how Galois theory was used to show that you can't construct an angle trisection helped you to understand what Galois theory is on a deeper level? I guess what i'm really getting at (as someone who is currently independently studying Galois theory) is Galois theory not to be completely understood from a purely algebraic viewpoint? I thought it was an algebraic topic with applications to other fields, but you seem to be saying that understanding the applications of Galois theory (i.e. disproving a general solution for a quintic or constructing an angle trisection) are KEY to a true understanding of Galois theory. Am I understanding you correctly?

Edit: I hope this thread keeps going, now that OP's question has been clarified it's getting even better! PF is so epic.

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
First of all, thank you for your whole post, i found it very insightful. Secondly, when you say that the difficulty in understanding Galois theory is that you need to know more than just solving polynomials, do you mean to say that understanding how Galois theory was used to show that you can't construct an angle trisection helped you to understand what Galois theory is on a deeper level? I guess what i'm really getting at (as someone who is currently independently studying Galois theory) is Galois theory not to be completely understood from a purely algebraic viewpoint? I thought it was an algebraic topic with applications to other fields, but you seem to be saying that understanding the applications of Galois theory (i.e. disproving a general solution for a quintic or constructing an angle trisection) are KEY to a true understanding of Galois theory. Am I understanding you correctly?

Edit: I hope this thread keeps going, now that OP's question has been clarified it's getting even better! PF is so epic.
Galois theory is the study of mathematical entities known as groups. Although the theory is known by his name, many other mathematicians before and since Galois have studied groups and how they may be applied to the study of other areas of math in general. Galois was a talented mathematician in his own right, and his tragic death at such as early age robbed mathematics of a bright light and contributed in no small way to his modern reputation. The image of a young man furiously scribbling his last thoughts the night before his impending demise is one which contains much drama.

I have not personally studied that much abstract math, and I am not familiar with the basics of group theory. The famous Rubik's Cube puzzle of the 1980's was one illustration of how group theory could be used to solve a concrete problem.

fresh_42
Mentor
The applications of Galois theory: non solvability of higher order polynomials, no construction of a trisection or doubling a cube are trivial. They are at most a remark at its end. To understand Galois theory it's necessary to understand automorphism groups and their relation to field extensions. The mentioned applications aren't of much help.

I found this for the 6 degree polynomial:

https://en.wikipedia.org/wiki/Kampé_de_Fériet_function

Kampé de Fériet function

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And this is about 7 degree polynomials:

Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by superimposing continuous functions of two variables.Hilbert's 13th problem was the conjecture this was not possible in the general case for seventh-degree equations. Vladimir Arnold solved this in 1957, demonstrating that this was always possible. However, Arnold himself considered the genuine Hilbert problem to be whether the solutions of septics may be obtained by superimposing algebraic functions of two variables (the problem still being open).