2 Restricted Power Series
Recall formal power series over a set \(K\) in one variable, written \(K [ \! [ x ] \! ] \), are infinite sums
This can be generalized to multivariate formal power series as
where \(x^{(n)} = x_1^{n_1} \cdots x_m^{n_m}\) with \(n = \sum _{i = 1}^m n_i\), and the sum is taken over all tuples \((n) = (n_1,\dots ,n_m)\) of non-negative integers.
When \(K\) is a ring, it is not hard to see that power series over \(K\) form a ring, and we will often refer to them as the ring of power series over \(K\).
We want to introduce a convergence property on the terms \(a_n x^n\), to do this we need a norm. Towards this we recall the definition of a normed ring: we say a ring \(R\) is a normed ring if we can equip it with a norm.
We now restrict ourselves to normed rings \(R\) and define a subset of power series with a convergence property.
Let \(f\) be a power series over \(R\) in one variable, write \(f = \sum _{n=0}^\infty a_n x^n\), and let \(c\) be a real number. We say \(f\) is a restricted power series of parameter \(c\) over \(K\) if
It would be nice if restricted power series of weight \(c\) were a ring, that is if we added, or multiplied, two restricted power series the resulting power series would also be restricted of weight \(c\).
If the norm of \(R\) has the non-archimedean property, it indeed turns out this is true. The non-archimedean property is key here; restricted power series form an additive group regardless of the norm on \(K\), but a stronger bound than the triangle inequality is necessary for the multiplication to be closed. This can be seen by the following example:
Let \(f = \sum _{n=0}^\infty \frac{(-1)^n}{\sqrt{n+1}}x^n\) be a power series over \(\mathbb {R}\) equipped with the standard norm. Then it is trivial to see that \(f\) is a restricted power series for the parameter 1. However, the product \(f^2\) has coefficients
and it can be shown that
Therefore \(\lim _{n \to \infty } \lvert b_n \rvert \geq \lim _{n \to \infty } \frac{2(n+1)}{(n+2)}=2\), that is \(f^2\) is not a restricted power series for parameter 1, and we see multiplication is not closed.
Restricted power series over a normed ring \(R\) form an additive group.
Restricted power series over a non-archimedean normed ring \(R\) form a ring.
We now suppose that \(K\) is a non-archimedean complete field; that is a field equipped with a metric that is complete and whose metric has the non-archimedean property as defined earlier, like \(\mathbb {Q}_p\).
When \(K\) is a non-archimedean complete field we denote the ring of restricted power series with parameter \(c\) over \(K\) by \(\mathcal{A}_c(K)\) and call it the Tate algebra over \(K\) of parameter \(c\).