Roots of polynomials as nonlinear systems of equations

[galoisjr]

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}$ (1)

Given some polynomial, the coefficients are known. Without the loss of generality, we can force the coefficient of the highest term to be zero, since we can always divide by it.

Next, we examine the same polynomial in it's factored form, which, from the FTOA, must exist and be equal to (1):

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}=\prod^{n}_{i=1}(x-x_{i})$ (2)

Now, we can expand the product and rewrite it:

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}=\sum^{n}_{k=0}(-1)^{n}e_{k}(x_{1},x_{2},...,x_{n})x^{n-k}$

Where the

$e_{k}(x_{1},x_{2},...,x_{n})=\sum_{1\leq j_{1}<j_{2}<...<j_{k}\leq n}x_{j_{1}}x_{j_{2}}\cdots x_{j_{k}}$

are the elementary symmetric polynomials. (If this looks odd to you the coefficient of the n-k power is the sum over the permutations of the n roots taken k at a time)

So, equating the known coefficients of the like powers of x, we have, a system of n nonlinear equations with n unknowns:

$a_{n}=1$
$a_{n-1}=\sum_{1\leq j_{1}\leq n}x_{j_{1}}=-(x_{1}+x_{2}+...+x_{n})$
$a_{n-2}=\sum_{1\leq j_{1}<j_{2}\leq n}x_{j_{1}}x_{j_{2}}=x_{1}x_{2}+...+x_{1}x_{n}+x_{2}x_{3}+...+x_{2}x_{n}+...+x_{n-1}x_{n}$

and so on.


Now, I am assuming that for n>4 the system of nonlinear equations is unsolvable, which would make sense because of galois theory and the obvious relation between the symmetric polynomials and the symmetric group on n letters. However, I have never read an algebra book that goes into the analysis as I just have here. I've seen some that mention permutation of coefficients, but not the actual roots. So, I was wondering if anyone has any suggestions on where to go from here from an abstract algebra viewpoint, and hoping someone can recommend a good book on nonlinear algebra.

 

[DonAntonio]

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}$ (1)

Given some polynomial, the coefficients are known. Without the loss of generality, we can force the coefficient of the highest term to be zero, since we can always divide by it.

Next, we examine the same polynomial in it's factored form, which, from the FTOA, must exist and be equal to (1):

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}=\prod^{n}_{i=1}(x-x_{i})$ (2)

Now, we can expand the product and rewrite it:

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}=\sum^{n}_{k=0}(-1)^{n}e_{k}(x_{1},x_{2},...,x_{n})x^{n-k}$

Where the

$e_{k}(x_{1},x_{2},...,x_{n})=\sum_{1\leq j_{1}<j_{2}<...<j_{k}\leq n}x_{j_{1}}x_{j_{2}}\cdots x_{j_{k}}$

are the elementary symmetric polynomials. (If this looks odd to you the coefficient of the n-k power is the sum over the permutations of the n roots taken k at a time)

So, equating the known coefficients of the like powers of x, we have, a system of n nonlinear equations with n unknowns:

$a_{n}=1$
$a_{n-1}=\sum_{1\leq j_{1}\leq n}x_{j_{1}}=-(x_{1}+x_{2}+...+x_{n})$
$a_{n-2}=\sum_{1\leq j_{1}<j_{2}\leq n}x_{j_{1}}x_{j_{2}}=x_{1}x_{2}+...+x_{1}x_{n}+x_{2}x_{3}+...+x_{2}x_{n}+...+x_{n-1}x_{n}$

and so on.


Now, I am assuming that for n>4 the system of nonlinear equations is unsolvable, which would make sense because of galois theory and the obvious relation between the symmetric polynomials and the symmetric group on n letters. However, I have never read an algebra book that goes into the analysis as I just have here.

Any medium-decent book in algebra that deals with fields extensions/Galois Theory talks of the above. You can check under

"symmetric polynomials or symmetric rational functions" the books by Hungerford, Lang, Dummit & Foote (Gallian is an exception!), Fraleigh, etc.

I've seen some that mention permutation of coefficients, but not the actual roots. So, I was wondering if anyone has any suggestions on where to go from here from an abstract algebra viewpoint, and hoping someone can recommend a good book on nonlinear algebra.

 

[HallsofIvy]

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial

$P(x)=\sum^{n}_{k=0}a_{k}x^{k}$ (1)

Given some polynomial, the coefficients are known. Without the loss of generality, we can force the coefficient of the highest term to be zero, since we can always divide by it.

Typo- you clearly mean "we can force the coefficient of the highest term to be one", not zero.

 

[galoisjr]

Thank you Don Antonio. And you're right. The book that my class used in college just mentioned them. However, I did read a little of Artin's book and I don't remember seeing anything about them, but I most likely overlooked it. I really enjoyed the book by Rotman. It is a very well organized book in my opinion and it's also written very concisely and to the point even though its 1000 pages. That's where I found information on them. And he actually makes it a point to say that "no classical formula analogous to the quadratic formula for the quintic exists, we did not say that no solution to the system exists." Which answered my question.

And thank you Halls of Ivy, I'll edit that. Obviously there is no way I would have made it this far into my degree if I thought that I could force a coefficient to be zero and then divide by it... shaking my head.

 

[blue_raver22]

I find this approach to examining polynomials as nonlinear systems of equations to be very interesting and potentially useful. It allows for a deeper understanding of the structure of polynomials and their roots, and may have applications in various fields such as mathematics, physics, and engineering.

One potential direction to explore would be to study the behavior of these nonlinear systems of equations as the degree of the polynomial increases. This could provide insights into the solvability of these systems and the behavior of the roots as the polynomial becomes more complex.

I would also suggest looking into the use of computer algebra systems or numerical methods to solve these systems of equations and analyze the behavior of the roots. This could provide a more concrete understanding and visualization of the relationships between the coefficients and roots of polynomials.

In terms of resources, I would recommend looking into books on abstract algebra or nonlinear algebra such as "Abstract Algebra" by David S. Dummit and Richard M. Foote, and "Nonlinear Algebra" by Bernd Sturmfels. These books provide a theoretical and mathematical foundation for studying nonlinear systems of equations and their applications.

Overall, this approach to examining polynomials as nonlinear systems of equations is a fascinating topic that has potential for further research and applications. I hope this response has been helpful and I wish you all the best in your studies.