sympyでsymmetrizeを調べた。

以下、汎用性はありません。

from sympy import *
var('x,y'  ,real=True)
var('s1,s2',real=True)
eq=Eq(x**3-x*y+y**3,1)
print("#",           eq.lhs )
print("#",symmetrize(eq.lhs))
print("#",symmetrize(eq.lhs,formal=True))
s=symmetrize(eq.lhs,formal=True)
print("#0  ",s[0])
print("#1  ",s[1])
print("#2  ",s[2])
print("#20 ",s[2][0])
print("#21 ",s[2][1])
print("#200",s[2][0][0])
print("#201",s[2][0][1])
print("#210",s[2][1][0])
print("#211",s[2][1][1])
print()
def mySymmetrize(myLhs):
    s=symmetrize(myLhs,formal=True)
    mySubs={s[2][0][0]:s[2][0][1],s[2][1][0]:s[2][1][1]}
    return s[0].subs(mySubs).expand()
print("#",mySymmetrize(eq.lhs))

# x**3 - x*y + y**3
# (-3*x*y*(x + y) - x*y + (x + y)**3, 0)
# (s1**3 - 3*s1*s2 - s2, 0, [(s1, x + y), (s2, x*y)])
#0   s1**3 - 3*s1*s2 - s2
#1   0
#2   [(s1, x + y), (s2, x*y)]
#20  (s1, x + y)
#21  (s2, x*y)
#200 s1
#201 x + y
#210 s2
#211 x*y

# x**3 - x*y + y**3

参考

問題1

a+b+c,a^2+b^2+c^2

var('a b c')
print("#",symmetrize(a**2+b**2+c**2))
# (-2*a*b - 2*a*c - 2*b*c + (a + b + c)**2, 0)

問題2

a+b+c,a3+b3+c3-3abc

参考

未整理