# ag.algebraic geometry – How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series?

How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series ?

We know that if $$f(x)=sum_{i=0}^{infty} a_ix^i$$ be a power series over $$p$$-adic field, then the Newton polygon of $$f(x)$$ is the lower convex hull of the segments joining $$(i, v(a_i))$$, where $$v$$ is the $$p$$-adic valuation.

Now consider the p-adic power series $$f(x_1,~x_2, cdots, x_n)=sum a_{h_i,~h_2, cdots, h_n} x_1^{h_1}x_2^{h_2} cdots x_n^{h_n}$$ of $$n$$-variables.

Question:

How to find Newton polygon of it in the above sense ?

My Intuition:

Although, I don’t have any concrete method but I have thought the following intuitions.

$$(1)$$ I not sure whether we can consider only one variable, say, $$x_i$$, and ignore the rest variablesin order to find the Newton polygon of $$f(x_1,cdots, x_n)$$.

But I am thinking about the following method:

$$(2)$$

$$(a)$$ We know that $$text{Newton polygon}$$ of product of two single variable power series is obtained by $$text{gluing}$$ together the segments of the $$text{Newton polygon}$$ of each individual power series under $$text{slope condition}$$. For example, if $$f(x)$$ and $$g(x)$$ be two p-adic power series. Let $$S(f)$$, $$S(g)$$ be respectively denote the slope sets of the segments (or straight lines) of Newton polygon of $$f(x)$$ and $$g(x)$$. Assume $$S(f) leq S(g)$$, then $$text{Newton polygon}$$ of the product $$f(x)g(x)$$ is obtained by $$text{gluing}$$ together the segments of $$text{Newton polygon}$$ of $$f(x)$$ and $$g(x)$$.

$$(b)$$ So we can extend this above method for finite product of power series.

$$(c)$$ Now if we assume that $$f(x_1,~x_2, cdots, x_n)=f_1(x_1) cdot f_2(x_2) cdots f_n(x_n)$$, then we can find $$text{Newton polygon}$$ of each factors $$f_i(x_i)$$ and by the above method, joining all segments of $$text{Newton polygon}$$ of each factor $$f_i(x_i)$$,we will get the Newton polygon of the multi-variable power series $$f(x_1,~x_2, cdots, x_n)$$.

Am I thinking right way ?