Efficient reduction to elementary symmetric polynomials

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I wasn't asking how to reduce the expression in elementary symmetric polynomials, but rather how to speed up the computation of SymmetricFunctions.from_polynomial given the special conditions I have. That is my input is some polynomial in the roots (which I know is symmetric) and I want to express it in $e_i$. But I guess there really isn't a way other than writing the reduction algorithm myself.

It's unclear what is your set up. If you work with symmetric expressions of 5 roots only, how do you get $e_i$ with $i>5$ in the result of from_polynomial? In general you can reduce your polynomial in the ideal generated by polynomials like $e_1$ (known to be zero) before converting to a symmetric one.

If you work with symmetric expressions of 5 roots only, how do you get ei with i>5 in the result of from_polynomial

The same as in your question you linked: My polynomial is of higher degree than the number of variables (which in my case is the hypothetical roots of a (different) polynomial to be solved)

you can reduce your polynomial in the ideal generated by polynomials like e1 (known to be zero) before converting to a symmetric one.

Yeah, that's what I tried at first but the reduce operation breaks the symmetry (quite naturally). E.g. (a^2 + b^2 + c^2 + d^2).reduce([a+b+c+d]) is no longer symmetric because the a is eliminated by the division algorithm,

I do not follow. (a^2 + b^2 + c^2 + d^2).reduce([a+b+c+d])is symmetric in the remaining three variables. E.g. see https://sagecell.sagemath.org/?q=jxrjub