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)$ .

**Answer:**

$ (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)$ .