Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial(adsbygoogle = window.adsbygoogle || []).push({});

[itex]P(x)=\sum^{n}_{k=0}a_{k}x^{k}[/itex] (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):

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

Now, we can expand the product and rewrite it:

[itex]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}[/itex]

Where the

[itex]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}}[/itex]

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:

[itex]a_{n}=1[/itex]

[itex]a_{n-1}=\sum_{1\leq j_{1}\leq n}x_{j_{1}}=-(x_{1}+x_{2}+...+x_{n})[/itex]

[itex]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}[/itex]

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.

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# Roots of polynomials as nonlinear systems of equations

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