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I'm reading about symmetric polynomials at the moment, and came upon this statement:

Let $G(x_1, \ldots, x_n)$ be a symmetric polynomial. Separate out $x_n$, representing $G$ as follows: $G_\nu x_n^\nu + \cdots+ G_1 x_n + G_0$, where the $G_i$ are polynomials in $x_1, \ldots, x_{n-1}$.

So far I understand why this is possible and what this means. But now the author claims that because $G$ is symmetric, so is each $G_i$. I don't see that. And in thinking about it, I realized that I don't really know what a symmetric polynomial is. I get that it's a polynomial in which after permuting the variables, you get "the same" polynomial, but what does "the same" mean? I can see at least two interpretations of "the same" for multivariable polynomials:

  1. The two polynomials represent the same function.
  2. One polynomial can be turned into the other using high-school algebra.

Of these two, the first seems more natural to me, because I've always thought of (2) as being justified because it implies (1). So (2) seems like a more round-about way of saying (1), and in fact (2) might even be strictly weaker than (1) - distributivity and so on just happen to be the rules taught to us in middle school, but who's to say there aren't polynomials equal for all values of their inputs that can't be transformed into one another by those manipulations?

Thus, what exactly is the author claiming here and how do you prove it?

Michael Hardy
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Jack M
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  • What a symmetric polynomial is, that is clear - http://en.wikipedia.org/wiki/Symmetric_polynomial. The examples given there may be helpful. – Dietrich Burde Mar 06 '15 at 11:23
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    over finite fields, distinct polynomials can represent the same function. Have you had a formal definition of what a polynomial is ? of what $K[x]$ is ? – mercio Mar 06 '15 at 11:31
  • @mercio I've seen $K[x]$ defined as a sequence of coefficients, I suppose, but that always struck me as an ugly definition. – Jack M Mar 06 '15 at 11:49
  • well then two sequences of coefficients are equal when ... um .. they are .. equal, I really don't know how to say this. (though it's a bit more messy when there are several variables involved) – mercio Mar 06 '15 at 11:51
  • @mercio It connects to what I said about point (2) in my question - it's not so much that I don't understand the definition as that I don't like it. Is there no more elegant definition? – Jack M Mar 06 '15 at 12:06

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$5x_1^2 x_2$ is not symmetric because interchanging $x_1$ and $x_2$ yields $5x_2^2 x_1$, which is a different polynomial.

$5x_1^2 x_2 + 4x_2^2 x_1$ is similarly not symmetric.

$5x_1^2x_2 + 5x_2^2x_1$ is symmetric because interchanging $x_1$ and $x_2$ yields $5x_2^2 x_1 + 5x_1^2x_2$, which is the same polynomial.

"The same polynomial" simply means that for every two monomials that are the same except for the coefficient, the coefficient is also the same.

If you're working in something like the integers modulo $7$, or any of various other rings or fields, one sometimes has different polynomials both representing the same function, so merely saying that permutation of variables gives you another polynomial representing the same function may be weaker than saying permutation of the variables gives you the same polynomial. Only if the latter holds is it is a symmetric polynomial. (I said "may be" because I don't have a good example involving at least two variables at the tip of my tongue.)

Michael Hardy
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