Exercise 2.2.14

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}}

We define the weight of $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ to be $a_1+2a_2+\cdots+na_n$.
\begin{enumerate}
\item[(a)] Prove that $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ is homogeneous and that its weight is the same as its total degree when considered as a polynomial in $x_1,\ldots,x_n$.
\item[(b)] Let $f = F(x_1,\ldots,x_n]$ be symmetric and homogeneous of total degree $d$. Show that $f$ is a linear combination of products $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ of weight $d$.
\end{enumerate}

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}
\begin{enumerate}
\item[(a)] By Ex. 2.2.13, each $\sigma_k$ being homogeneous of degree $k$, the product $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ is homogeneous. As $\deg(\sigma_k) = k$, $\deg(\sigma_1^{a_1}\cdots \sigma_n^{a_n}) = a_1+2a_2+\cdots+na_n$ is equal to the weight of $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$.
\item[(b)] Since $f$ is symmetric, $f$ is a linear combination of products $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$. These products being homogeneous of degree $a_1+2a_2+\cdots+na_n$, and $f$ being homogeneous, by Ex 2.2.13(b), each term of this sum has degree $d$.

Conclusion : $f$ is a linear combination of products $\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ of weight $d$.

\end{enumerate}
\end{proof}

Exercise 2.2.14

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Exercise 2.2.14

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