Exercise 2.2.15

Proof.

(a)

If $x_1$ appears in a term $c{x}_1^{a_1}{x}_2^{a_2}\cdots x_n^{a_n}$ of $f$, then the transposition $\tau =(1,2)$ applied to $f$ show that $c{x}_2^{a_1}{x}_1^{a_2}\cdots x_n^{a_n}$ is a term of $f$, so $x_2$ appears in a term of $f$ with the same exponent. Thus the maximal exponent is the same for the two variables : $${\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_1(f)={\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_2(f),$$

and the same is true for any pair of variables.

(b)

As ${\sigma }_k=\underset{1\le i_1<\cdots <i_k\le n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{x}_{i_1}{x}_{i_2}\cdots x_{i_k}$, ${\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i({\sigma }_k)=1$. For polynomial of one variable $x$, $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{pq})=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)+\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(q)$, and ${\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_1(f)$ is the degree in $x_1$ of $f$ as an element of $k[x_2,\dots ,x_n][x_1]$, so $${\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i(\mathit{fg})={\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i(f)+{\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i(g).$$

Therefore ${\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i({\sigma }_1^{a_1}\cdots {\sigma }_n^{a_n})=a_1{\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i({\sigma }_1)+\cdots +a_n{\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->}_i({\sigma }_n)=a_1+\cdots +a_n$.