# I got stuch finding Newton polygon of the following product with any easiest method

Find the Newton polygon of the following polynomials:

$$(i) \ f(X)=(1-X)(1-pX)(1-p^3X)$$,

$$(ii) \ g(X)=\prod_{i=1}^{p^2} (1-iX)$$.

$$(i)$$

To find the Newton polygon for the polynomial $$f(X)$$, we can simply multiply the linear factors as follows:

$$f(X)=(1-X)(1-pX)(1-p^3X) \\ \Rightarrow f(X)=1-(p^3+p+1)X+(p^4+p^3+p)X^2-p^4 X^3$$.

Thus the Newton polygon has following vertices:

$$(0, ord_p(1)), \ (1, ord_p(-p^3+p+1)), \ (2, ord_p(p^4+p^3+p)) , \ (3,ord_p(p^4))$$

i.e., $$(0,0), \ (1, 0), \ (2,1), \ (3,4)$$.

But this process we can not apply to the second polynomial $$g(X)$$ in (ii) because it will be complicated.

So there should be other easy method to find the Newton polygon for the polynomials in $$(i)$$ and $$(ii)$$.

Please help me find the Newton polygon for $$(ii)$$.