A univariate polynomial
(1)
has degree (or grading) , the largest with .
Every polynomial ring is a graded ring (see e.g. wiki) with a monimial
(2)
grading defined as total degree .
Symmetric sums 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
(3)
can potentilly lower the grading of test function by , thus it's assigned grading . By the same reasoning, operator
(4)
has grading .