Exercise 2.3.1

Proof. We repeat Examples 2.3.1 and 2.3.2 with Sage :

$\text{∙}$ We build the Groebner basis of the ideal $⟨e_1-y_1,e_2-y_2,e_3-y_3⟩$, where $e_1,e_2,e_3$ are the elementary symmetric polynomials in $x_0,x_1,x_2$ :

e = SymmetricFunctions(QQ).e()
e1, e2, e3 = e([1]).expand(3),e([2]).expand(3),e([3]).expand(3)
R.<x0,x1,x2,y1,y2,y3> = PolynomialRing(QQ, order = ’degrevlex’)
J = R.ideal(e1-y1, e2-y2, e3-y3)
G = J.groebner_basis()

$\text{∙}$ We compute the coefficients of $f=(x-x_0^3)(x-x_1^3)(x-x_2^3)$ as polynomials in $x_1,x_2,x_3$:

f = (x-x0^3) * (x-x1^3) * (x-x2^3)
coeffs = f.coefficients(x, sparse = False)
coeffs = map(lambda c : R(c), coeffs)
coeffs

$\text{∙}$ The same coefficients as polynomials in ${\sigma }_1,{\sigma }_2,{\sigma }_3$:

var(’sigma_1,sigma_2,sigma_3’)
ncoeffs = [c.reduce(G) for c in coeffs]
nncoeffs = [c.subs(y1 = sigma_1,y2 = sigma_2,y3 = sigma_3) for c in ncoeffs]
nncoeffs

$\text{∙}$ We apply the substitution ${\sigma }_1\mapsto -2,{\sigma }_2\mapsto -3,{\sigma }_3\mapsto -5$ and compute the polynomial $p$ whose roots are ${\alpha }_1^3,{\alpha }_2^3,{\alpha }_3^3$, where ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are the roots of $y^3+2y^2-3y+5$.

nncoeffs = [c.subs(sigma_1 = -2, sigma_2 = -3, sigma_3 = -5) for c in nncoeffs]
p = sum(nncoeffs[i]*y^i for i in range(1+f.degree(x)))
p

$\text{∙}$ Numerical verification:

S.<y>  = PolynomialRing(ComplexField(prec = 40))
[c[0] for c in S(p).roots()]

q = y^3+2*y^2-3*y+5
l = [c[0]^3 for c in q.roots()]
l

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