symmetric monoidal (∞,1)-category of spectra
A power series in a variable $X$ and with coefficients in a ring $R$ is a series of the form
where $a_n$ is in $R$ for each $n\ge 0$. Given that there are no additional convergence conditions, a power series is also termed emphatically as a formal power series. If $R$ is commutative, then the collection of formal power series in a variable $X$ with coefficients in $R$ forms a commutative ring denoted by $R [ [ X ] ]$.
More generally, a power series in $k$ commuting variables $X_1,\ldots, X_k$ with coefficients in a ring $R$ has the form $\sum_{n_1=0,n_2=0,\ldots, n_k = 0}^\infty a_{n_1\ldots n_k} X_1^{n_1} X_2^{n_2}\cdots X_k^{n_k}$. If $R$ is commutative, then the collection of formal power series in $k$ commuting variables $X_1,\ldots, X_k$ form a formal power series ring denoted by $R [ [ X_1,\ldots, X_k ] ]$.
More generally, we can consider noncommutative (associative unital) ring $R$ and words in noncommutative variables $X_1,\ldots, X_k$ of the form
(where $m$ has nothing to do with $k$) and with coefficient $a_w \in R$ (here $w$ is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form $\sum_w a_w X_w$ and they form a formal power series ring in variables $X_1,\ldots, X_k$ denoted by $R\langle \langle X_1,\ldots, X_k \rangle\rangle$. Furthermore, $R$ can be even a noncommutative semiring in which case the words belong to the free monoid on the set $S = \{ X_1,\ldots, X_k\}$, the partial sums are then belong to a monoid semiring $R\langle S\rangle$. The formal power series then also form a semiring, by the multiplication rule
Of course, this implies that in a specialization, $b$-s commute with variables $X_{i_k}$; what is usually generalized to take some endomorphisms into an account (like at noncommutative polynomial level of partial sums where we get skew-polynomial rings, i.e. iterated Ore extensions).
Let $R$ be a commutative ring, and let $R[X]$ be the polynomial ring on one indeterminant $X$. Then $(X)$ is a maximal ideal in $R[X]$, and results in an adic topology on $R[X]$. The ring of formal power series in $R$ is the adic completion of the limit of the quotient of $R[X]$ by powers of $(X)$:
The ring of formal power series for multiple indeterminants $X_i$ is constructed iteratively: because $R[[X_1, X_2, \ldots X_n]]$ is a commutative ring, one could construct the polynomial ring $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ on the indeterminant $X_{n+1}$. As above, $(X_{n+1})$ is a maximal ideal in $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ with a corresponding adic topology, and one can then take the adic completion
The resulting commutative ring is usually just written as $R[[X_1, X_2, \ldots X_n, X_{n+1}]]$.
For a natural number $k$, a power series $\sum_{n=0}^\infty a_n X^n$ such that $a_n = 0$ for all $n \gt k$ is a polynomial of degree at most $k$.
For $f \in C^\infty(\mathbb{R})$ a smooth function on the real line, and for $f^{(n)} \in C^\infty(\mathbb{R})$ denoting its $n$th derivative its MacLaurin series (its Taylor series at $0$) is the power series
If this power series converges to $f$, then we say that $f$ is analytic.
This follows easily from the observation that we can invert $1 + x b$ for any power series $b$ by forming $1 - x b + x^2 b^2 - \ldots$ and collecting only finitely many terms in each degree. As a simple corollary,
$R[ [x_1, \ldots, x_n] ]$ equipped with the ideal $(x_1, \ldots, x_n)$ is the free adic $R$-algebra on $n$ generators, in the sense that it is the value of the left adjoint $Pow$ to the forgetful functor
as applied to the set $\{x_1, \ldots, x_n\}$.
The idea is that for each adic $R$-algebra $(S, I)$ and element $(s_1, \ldots s_n) \in I^n$, there is a unique adic algebra map $R[ [x_1, \ldots, x_n] ] \to S$ that sends $x_i$ to $s_i$; this adic algebra map sends a power series $\sum a_{k_1, \ldots, k_n} x_1^{k_1} x_n^{k_n}$ to the sequence of truncations
belonging to $\underset{\longleftarrow}{\lim}_k S/I^k \cong S$.
It follows that we may define a clone or cartesian operad as follows: the $n^{th}$ component is the set $I_n = (x_1, \ldots, x_n) \subset R[ [x_1, \ldots, x_n] ]$ which is the monad value $Ideal Pow(\{x_1, \ldots, x_n\})$. Letting $M$ denote the monad $Ideal \circ Pow$, with monad multiplication $\mu$, and $[n]$ the set $\{x_1, \ldots, x_n\}$, the clone multiplication
is the composition of the maps
The clone multiplication thus defined is called substitution of power series; it takes a tuple consisting of $p(x_1, \ldots, x_n) \in I_n, q_1(x_1, \ldots x_k) \in I_k, \ldots q_n(x_1, \ldots, x_k) \in I_k)$ to a power series denoted as
The resulting clone or operad yields, in the particular case $k = n = 1$, an associative substitution operation
with $sub(p, q) = p \circ q$ the power series $p(q(x))$.
The group of invertible elements in the substitution monoid $x R[ [x] ]$ consists of power series of the form $a_1 x + a_2 x^2 + \ldots$ where $a_1$ is multiplicatively invertible in the ring $R$.
In other words, we can functionally invert a power series provided that the linear coefficient $a_1$ is invertible in $R$.
Given power series $a = a_1 x + a_2 x^2 + \ldots$ and $b = b_1 x + b_2 x^2 + \ldots$, we may read off coefficients of the composite $a \circ b$ as
where in particular $(a \circ b)_1 = a_1 b_1$. Now $a$ is the left functional inverse of $b$, or $b$ is the right inverse of $a$, if $(a \circ b)(x) = x$, i.e., if $(a \circ b)_k = 1$ if $k = 1$ and $0$ otherwise. The first equation says simply $(a \circ b)_1 = a_1 b_1 = 1$ which implies $a_1$ is invertible. Conversely, if $a_1$ is multiplicatively invertible and $b_1 = a_1^{-1}$, then the equations
may be uniquely solved for the remaining $a_i$'s given the $b_j$'s, and uniquely solved for the remaining $b_j$'s given the $a_i$'s, by an inductive procedure: for $k \neq 1$ we have
and this allows us to solve for $b_k$,
given the values $a_1, \ldots, a_k$ and earlier $b$-values $b_{k_j}$ for $k_j \lt k$ given by inductive hypothesis. Similarly we can solve for $a_k$ in terms of given coefficients $b_1, \ldots, b_k$ and earlier $a$-values $a_n$, $n \lt k$. Thus every power series $a$ has a right inverse if $a_1^{-1}$ exists, and $b$ has a left inverse if $b_1^{-1}$ exists, and this completes the proof.
One way to define the formal differentiation operator, as a function $\frac{\partial}{\partial X}:R[[X]] \to R[[X]]$, is via the usual formula
Then $\frac{\partial}{\partial X}$ is an $R$-linear function on $R[[X]]$ which satisfies the Leibniz rule, meaning that it is a derivation and $R[[X]]$ is a differential algebra.
Here is a conceptual story underlying the formalism. Let $D = R[\varepsilon]/(\varepsilon^2)$ be the representing object for derivations (the “ring of dual numbers”). Let $\delta: R[ [X] ] \to R[ [X] ] \otimes_R D \cong R[ [X] ][\varepsilon]/(\varepsilon^2)$ be the unique topological $R$-algebra map (under the $(X)$-adic topologies described above) that sends $X$ to $X + \varepsilon$. (If it helps, think $\delta(q) = q(X + \varepsilon)$.)
For $p \in R[ [X] ]$, the derivative $p'$ is the unique element of $R[ [X] ]$ satisfying
We leave as an exercise the proof that the two definitions of derivative match:
(Hint: the restriction of $p \mapsto p'$ to $R[X]$ is by construction a derivation such that $X' = 1$, and $(X^k)' = k X^{k-1}$ by induction. This induces derivations on quotient algebras $R[X]/(X^n)$, satisfying the same formula. Then pass to the inverse limit.)
(Chain rule) For $p \in R[ [X] ]$ and $q \in x R[ [X ] ]$,
See here for a conceptual proof, using the universal property of adic completion.
Relatedly but in a slightly different direction, we can consider differentiation in coalgebraic terms. Suppose the commutative ring $R$ is a commutative algebra over $\mathbb{Q}$ (thus permitting division by nonzero integers). Then the set $R[ [X]]$ may be identified with the terminal coalgebra $R^\mathbb{N}$ of the endofunctor $R \times - \colon Set \to Set$ via the map
whereby the coalgebra structure on $R^\mathbb{N}$,
corresponds to
One may then apply coinductive techniques to prove various facts. One illustration is given here, where coinduction on power series is used to prove the general binomial theorem
where, remarkably, $r$ is an arbitrary element of $R$.
ring with infinitesimals | function |
---|---|
dual numbers | differentiable function |
Weil ring | smooth function |
power series ring | analytic function |
Suppose that $K$ is a Archimedean ordered field and $K[[\epsilon]]$ is the ring of power series in $K$. Since $K[[\epsilon]]$ is a local ring, the quotient of $K[[\epsilon]]$ by its ideal of non-invertible elements $\epsilon K[[\epsilon]]$ is the residue field $K$ itself, and the canonical function used in defining the quotient is the function $\Re:K[[\epsilon]] \to K$ which takes a number $a \in K[[\epsilon]]$ to its purely real component $\Re(a) \in K$ and takes $\Re(\epsilon) = 0$. Since $K[[\epsilon]]$ is an ordered $K$-algebra, there is a strictly monotone ring homomorphism $h:K \to K[[\epsilon]]$. An element $a \in K[[\epsilon]]$ is purely real if $h(\Re(a)) = a$, and an element $a \in K[[\epsilon]]$ is purely infinitesimal if it is in the fiber of $\Re$ at $0 \in K$. Zero is the only element in $K[[\epsilon]]$ which is both purely real and purely infinitesimal.
Suppose that $K$ is a sequentially Cauchy complete Archimedean ordered field with lattice structure, and $K[[\epsilon]]$ is the ring of power series of $K$. Then analytic functions are each definable on $K$ using the algebraic, order, metric, and convergence structure on $K$.
The ring homomorphism $h:K \to K[[\epsilon]]$ preserves analytic functions: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\eta \in \epsilon K[[\epsilon]]$, then for every analytic function $f \in C^\infty(K)$, there is a function $f_{K[[\epsilon]]}:K[[\epsilon]] \to K[[\epsilon]]$ such that for all elements $x \in K$, $f_{K[[\epsilon]]}(h(x)) = h(f(x))$ and
A formalization in homotopy type theory and there in Coq is discussed in section 4 of
The discussion of the differentiation of a converging power series term by term is at
Last revised on May 28, 2023 at 20:20:06. See the history of this page for a list of all contributions to it.