Exercise 2.2.17

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Use the Newton identities (2.22) to express the power sum $s_2,s_3,s_4$ in terms of the elementary symmetric polynomials $\sigma_1,\sigma_2,\sigma_3,\sigma_4$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
$s_r = x_1^r+x_2^r+\cdots+x_n^r$.

We suppose here that the number $n$  of variables is at least 4.
Then $$s_r = \sigma_1 s_{r-1}-\sigma_2 s_{r-2}+\cdots+(-1)^r\sigma_{r-1}s_1 + (-1)^{r-1} r \sigma_r.$$
\begin{align*}
s_1 &= \sigma_1,\
\
s_2 &= \sigma_1 s_1 - 2 \sigma_2\
&= \sigma_1^2-2\sigma_2,\
\
s_3 &= \sigma_1 s_2 -  \sigma_2s_1 +3\sigma_3\
&=\sigma_1(\sigma_1^2 - 2 \sigma_2) - \sigma_2 \sigma_1 + 3 \sigma_3\
&=\sigma_1^3-3\sigma_1\sigma_2+3\sigma_3,\
\
s_4 &= \sigma_1s_3 - \sigma_2 s_2 + \sigma_3 s_1 - 4 \sigma_4\
&=\sigma_1(\sigma_1^3-3\sigma_1\sigma_2+3\sigma_3) - \sigma_2(\sigma_1^2-2\sigma_2)+ \sigma_3\sigma_1 - 4 \sigma_4\
&=\sigma_1^4-4\sigma_1^2\sigma_2+4\sigma_1\sigma_3+2\sigma_2^2-4\sigma_4.
\end{align*}

Verification with Sage:

\begin{verbatim}
e = SymmetricFunctions(QQ).e()
e1, e2, e3, e4 = e([1]).expand(4),e([2]).expand(4),e([3]).expand(4),e([4]).expand(4)
R.<x0,x1,x2,x3,y1,y2,y3,y4> = PolynomialRing(QQ, order = 'lex')
J = R.ideal(e1-y1,e2-y2,e3-y3,e4-y4)
G = J.groebner_basis()
s2 = x0^2 + x1^2 + x2^2 + x3^2
s3 = x0^3 + x1^3 + x2^3 + x3^3
s4 = x0^4 + x1^4 + x2^4 + x3^4
g2, g3, g4 = s2.reduce(G),s3.reduce(G),s4.reduce(G)
var('sigma_1,sigma_2,sigma_3,sigma_4')
h2 = g2.subs(y1=sigma_1,y2=sigma_2,y3=sigma_3,y4=sigma_4)
h3 = g3.subs(y1=sigma_1,y2=sigma_2,y3=sigma_3,y4=sigma_4)
h4 = g4.subs(y1=sigma_1,y2=sigma_2,y3=sigma_3,y4=sigma_4)
\end{verbatim}

\begin{verbatim}
h2, h3, h4
\end{verbatim}
$$\left(\sigma_{1}^{2} - 2 \, \sigma_{2}, \sigma_{1}^{3} - 3 \, \sigma_{1}
\sigma_{2} + 3 \, \sigma_{3}, \sigma_{1}^{4} - 4 \, \sigma_{1}^{2}
\sigma_{2} + 2 \, \sigma_{2}^{2} + 4 \, \sigma_{1} \sigma_{3} - 4 \,
\sigma_{4}ight).$$
\end{proof}

Exercise 2.2.17

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Exercise 2.2.17

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