How to convert a symmetric function into a polynomial on elementary symmetric functions?

SymmetricFunctions

asked 2024-09-21 08:49:53 +0200

Ys1123
5 ●4

By the Fundamental Theory of Symmetric Polynomials every symmetric polynomial in $\mathbb{C}[x_1,…,x_n]$ can be written uniquely in the elementary symmetric functions $e_1,…,e_n$. How can we obtain the expression if we are given a concrete symmetric polynomial on sagemath?

I want to know the value $(x^2-x-2)(y^2-y-2)(z^2-z-2)$ when $x+y+z=0, xy+yz+zx=-3, xyz=-1$.

I asked ChatGPT and searched on asksage , but I was unable to finally find what I was looking for.

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The page I saw was for example : https://ask.sagemath.org/question/325...

Ys1123 ( 2024-09-21 08:50:27 +0200 )edit

First create a symmetric function from the polynomial, then convert it to the ebasis. See https://doc.sagemath.org/html/en/refe...

FrédéricC ( 2024-09-21 09:05:04 +0200 )edit

The link is too wide. Which command do you intend respectively?

Ys1123 ( 2024-09-21 09:57:40 +0200 )edit

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answered 2024-09-22 16:34:23 +0200

Max Alekseyev

6355 ●8 ●45 ●150

updated 2024-09-22 16:37:31 +0200

Same idea as in @FrédéricC's answer applied to the OP's example:

I want to know the value $(x^2-x-2)(y^2-y-2)(z^2-z-2)$ when $x+y+z=0, xy+yz+zx=-3, xyz=-1$.

e_value = [1, 0, -3, -1]     # given values for e[i], i=0..3
e = SymmetricFunctions(QQ).e()
x, y, z = polygens(QQ, 'x,y,z')
f = e.from_polynomial( (x^2-x-2)*(y^2-y-2)*(z^2-z-2) )
v = sum( c * prod(e_value[ti] if ti<len(e_value) else 0 for ti in t) for t,c in f )
print(v)

It gives value 9.

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answered 2024-09-21 13:13:41 +0200

FrédéricC

5186 ●3 ●43 ●113

Like this

sage: e = SymmetricFunctions(QQ).e()
sage: x, y, z = polygens(QQ, 'x,y,z')
sage: f=1+(x+y+z)**3
sage: e.from_polynomial(f)
e[] + e[1, 1, 1]

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Thank you I checked it works. And I got thate[a] means elementary symmetric polynomial of degree a ande[a,b,c,…] means the producte[a]*e[b]*e[c]*… Then can you satisfy my second desire? (“ I want to know the value…”) I feel like it’s difficult owing to such wired notations.

Ys1123 ( 2024-09-22 12:34:41 +0200 )edit

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