When can we decompose a multivariable p-adic power series into product of single variable power series?

Question

Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?

For example, consider the following power series in $n$ variables:$$ f(x_1,~x_2, \cdots, x_n)=\sum_{j_1,~j_2,\cdots, j_n=0}^{\infty} a_{j_1,~j_2, \cdots, j_n} \prod_{k=1}^{n} (x_k-c_k)^{j_k}.$$

Now we want to express $f(x_1,~x_2, \cdots, x_n)$ in the following way: $$ f(x_1,~x_2, \cdots, x_n)=\left( \sum_{i_1=0}^{\infty} a_{i_1} x_1^{i_1}\right) \cdot \left( \sum_{i_2=0}^{\infty} a_{i_2} x_2^{i_2}\right) \cdots \left( \sum_{i_n=0}^{\infty} a_{i_n} x_n^{i_n}\right).$$ When and under which condition is it possible?

Is there any results or notes or resources available in this regard ?

Thanks,

Answer 1

Some night thoughts on your question which are too long for a comment. For simplicity, I will look at the two variable case. Firstly, there is a very simple discrete criterion for when a function of two variable splits in the way you are interested in: Let $f$ be a function from $X \times Y$. Then is can be represented as a product of two functions of one variable if and only if $$ f(x_1,y_1)f(x_2,y_2)=f(x_1,y_2)f(x_2,y_1) $$ for all possible values of the variables.

This is purely set theoretical situation but one can ask the same question in various contexts (continuous, smooth functions, or power series) and a small additional argument is required to show that if a function splits in the above sense, then the factors automatically have the required smoothness.

With respect to a differential condition, I will cheat and suppose that $f$ is a smooth real-valued function on the plane. (I know nothing about the $p$-adic case but hope that this might give you some pointers). Then, as above, $f f_{xy}=f_xf_y$ is a necessary condition for splitting and the question is whether it is sufficient. This is certainly true (using logarithms) if $f$ has no zeroes. In cases (like yours) where it can only have isolated zeroes I imagine that one could use a localisation argument to prove the sufficiency but I haven’t examined the details.