Symmetric PolynomialsDate: 07/31/97 at 05:43:03 From: Jillian Stewart Subject: Symmetric polynomials Hello! I have a question about polynomials and I hope you can help me (no one else could). How can it be proved that any symmetric polynomial can be expressed as an elementary symmetric polynomial? (I don't even know what symmetric polynomials are.) Thank you! Date: 07/31/97 at 11:36:48 From: Doctor Wilkinson Subject: Re: Symmetric polynomials A symmetric polynomial is one that doesn't change when you permute its variables. For example, x^2 + y^2 is a symmetric polynomial in x and y, because if you exchange x and y, the polynomial doesn't change; but x^2 + 2y^2 isn't symmetric, because if you exchange x and y you get 2x^2 + y^2, which is different. The elementary symmetric polynomials in x1, x2, x3, x4, ..., xn are s1 = x1 + x2 + x3 + ... + xn s2 = x1x2 + x1x3 +... +x2x3 + x2x4 +... + ... +... +xn-1xn ... sn = x1x2x3...xn That is, the kth symmetric polynomial is formed by taking all the possible products of k variables and adding them together. The theorem you cite is non-trivial but it is not really hard either. It can be proved by induction on the number of variables and you can find a proof in a lot of advanced algebra books. An example of what it is saying is that x^2 + y^2 = (x + y)^2 - 2xy = (s1(x, y))^2 - 2s2(x, y) If you try some other examples, you can sort of figure out how a general proof has to go, but you have to be pretty good at induction proofs to get it all to fit together. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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