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In Wikipedia's proof of the fundamental theorem of symmetric polynomials, it states that the proof focuses on the case where the polynomial is homogeneous, and that "The general case then follows by splitting an arbitrary symmetric polynomial into its homogeneous components".

What does this mean, and how can we accomplish this?

Jack M
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That is not strictly necessary, it is just a device to give the proof more structure. What you really need is to consider among the monomials of highest total degree (their sum forms one of the homogeneous components) the one with the highest lexicographic order.

This can then be reduced so that this specific degree combination is no longer present. Since the "glex" or graded lexicographic order is a monomial order, this degree combination will also not be reintroduced in later steps.

Lutz Lehmann
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  • So the decomposition into homogeneous parts just means grouping the monomials by degree? – Jack M Apr 22 '14 at 15:53
  • Yes. It only becomes complicated if you want to treat the polynomial as a black box where you then have to extract the components or monomials. The standard way to do that is to consider $p(t\cdot x)$ or $t^d\cdot p(x/t)$, with $x$ the vector of variables, and sort for the powers of $t$ or to compute the Taylor expansion for $t$, treating $x$ as parameters. – Lutz Lehmann Apr 22 '14 at 15:59