# Talk:Alternating polynomial

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Field:  Algebra

I removed the following:

If the characteristic of the coefficient ring is 2, [...] one generally defines it differently. (Vanishing if two entries are equal.)

I believe the author here confused alternating polynomials with alternating multilinear forms. The latter are indeed defined as vanishing if two entries are equal; if the characteristic is not 2, this is equivalent to a change in sign for any exchange of two arguments. But for alternating polynomials in characteristic 2 this definition is not useful: it creates a much weaker concept without much relation to the original one. For instance, the statement "the product of two alternating polynomials is symmetric" would not be true anymore. AxelBoldt (talk) 16:47, 19 December 2008 (UTC)

It is true that defining alternating as vanishing for two equal arguments is too weak, whether the characteristic is 2 or not: for instance ${\displaystyle x(x-y)}$ would be alternating (because the substitution y:=x kills the second factor), while its square ${\displaystyle x^{2}(x^{2}-2xy+y^{2})}$ is not even symmetric (in any characteristic). However setting "alternating=symmetric" in characteristic 2 is also too weak if one wants the theory to be even remotely similar to the case of other characteristics. For instance an alternating polynomial should not have any monomials with equal exponents (as xy in two variables does have) and in particular any monomial should involve at least all variables but one (because two variables with exponent 0 are forbidden). To me it makes no sense to consider for instance a constant polynomial, or the product of all variables, to be an alternating polynomial exclusively in characteristic 2. For one thing, this would make the definition of Schur polynomials unnatural in characteristic 2 (since the alternants would not span the space of alternating polynomials), while Schur polynomials enjoy the same properties there as in any other characteristic. There remains the problem of defining alternating polynomials in characteristic 2. I would suggest that the right definition is either "the linear span of the alternants" (where alternants are actually symmetric in characteristic 2, but that doesn't hurt), or as "the product of the Vandermonde polynomial and a symmetric polynomial". Both (equivalent) alternatives have a somewhat artificial flavour, but I think it is no shame to have definitions tailor-made to have nice consequences; I'd prefer this to more natural seeming definitions that allow no useful theory to be developed. But I'll try to look if I can find any definition in the books...Marc van Leeuwen (talk) 10:33, 3 November 2009 (UTC)

The footnote [2] and the remark preceding it are incorrect; witness switching x_1 and x_3, for example. I hope someone will be kind enough to update the article. Tesseran (talk) 17:10, 26 May 2009 (UTC)