A property of leading monomials in modular polynomial invariants

We study modular polynomial invariants of the cyclic group $C_p$ over a field of characteristic $p$ where $p$ is a prime number and use the reverse lexicographic order. We focus on the leading monomial of an invariant by considering the degrees of the terminal variables. It is obtained that this degree of each terminal variable is divisible by $p$ when only pure powers of terminal variables appear in the leading monomial. Then, we show that this divisibility also holds for the general case, that is, the degrees of the terminal variables of the leading monomial are divisible by $p$. After proving this property, we investigate the cyclic group $C_{p^k}$ for a positive integer $k$ with the same characteristic $p$. By noticing that the same arguments with only minor changes can be applied to this case, we get that $p$ divides the degree of each terminal variable.