Grading

A univariate polynomial

(1)

has degree (or grading) $n$ , the largest $i$ with $a_i \ne 0$ .

Every polynomial ring is a graded ring (see e.g. wiki) with a monimial

$\begin{equation} c_{\alpha_1,\dots, \alpha_n} x_1^{\alpha_1} \cdots x_n^{\alpha_n} \end{equation}$ (2)

grading defined as total degree $|\alpha| = \alpha_1 + \cdots + \alpha_n$ .

Symmetric sums $p_{\lambda}$ as a consequence have grading .

Operators on polynomial ring can both increase and decrease grading of elements they are acting on, thus the definition of grading is naturally extended to this case. Operator

$\begin{equation} \frac{\partial }{\partial p_{\lambda}} \end{equation}$ (3)

can potentilly lower the grading of test function by , thus it's assigned grading $-|\lambda|$ . By the same reasoning, operator

$\begin{equation} p_{\lambda} \frac{\partial }{\partial p_{\mu}} \end{equation}$ (4)

has grading $|\lambda| - |\mu|$ .