Algebra: it matters

I’m looking at two different models for learning polynomial functions, and trying to determine if they are equivalent. After a couple days of thinking, I’ve reduced the question to the following:

Can every symmetric polynomial of degree \(r\) in \(d\) variables that has no constant term be written as a sum of the \(r\)-th powers of linear polynomials in \(d\) degrees and a homogeneous polynomial of degree \(r\) each of whose monomials involves at most \(d-1\) variables?

  • Patrick Sanan

    Should it be “linear polynomials in d variables”?

  • Guest

    • Did you mean, raising a linear (combination of the variables) polynomial in d variables to the rth power?

    For example, (x1 + x2 + x3 + . . . + xd)^r ?

    Or did you mean summing all the variables raised to the rth power from linear polynomials in d variables?

    But if they are linear, then how there be an rth
    powered variable?

    • With or without coefficients?

    With, you can fudge an equality.

    So, you want either to add something raised to the rth power or sum over some things raised to the rth power with a
    non-determined number of monomials from a homogeneous polynomial which somehow is suppose to equal a symmetric polynomial, yes?

    An example of a symmetric polynomial given in Wikipedia is a counter-example to what you suspect.

    A full proof either way requires Galois Theory which is beyond my pay grade.

    Curious, this connexion of models for your learning process is just for you only, right?

  • sheldonross

    • Did you mean, raising a linear (combination of the variables) polynomial in d variables to the rth power?

    For example, (x1 + x2 + x3 + . . . + xd)r ?

    Or did you mean summing all the variables raised to the rth power from linear polynomials in d variables?

    But if they are linear, then how can there be an rth powered variable?

    • With or without coefficients?

    With, you can fudge an equality.

    So, you want either to add something raised to the rth power or sum over some things raised to the rth power with a
    non-determined number of monomials of a homogeneous polynomial which somehow is suppose to equal a symmetric polynomial, yes?

    An example of a symmetric polynomial given in Wikipedia is a counter-example to what you suspect.

    A full proof either way requires Galois Theory which is beyond my pay grade.

    Curious, this connexion of models for your learning process is just for you only, right?