Effective power series computations by Joris van der Hoeven CNRS, LIX, École polytechnique 91128 Palaiseau Cedex France Email: vdhoeven@lix.polytechnique.fr November 4, 2023

1Introduction

There are two main aspects about effective computations with formal power series. On the one hand, we need fast algorithms for the computation of coefficients. There is an important literature on this subject and the asymptotically fastest methods either rely on Newton's method [1] or on relaxed power series evaluation [7].

On the other hand, there is the problem of deciding whether a given power series is zero. This problem is undecidable in general, since we need to check the cancellation of an infinite number of coefficients. Therefore, a related subject is the isolation of sufficiently large classes of power series such that most of the common operations on power series can be carried out inside the class, but such that the class remains sufficiently restricted such that we can design effective zero tests.

In section 2, we first recall the most common operations on formal power series over a field of characteristic zero: the -algebra operations, restricted division, composition, the resolution of implicit equations, and so called restricted monomial transformations with arbitrary rational exponents. A subset of which is stable under each of these operations will be called a tribe. We will also specify effective counterparts of these notions.

The main result of this note is presented in section 4: given an effective tribe with an effective zero test, we show that the tribe also satisfies an effective version of the Weierstrass preparation theorem [10], and we give an algorithm for performing Weierstrass division with remainder. Our result can for instance be applied to the tribes of algebraic power series and D-algebraic power series (see also [4, 5, 9]).

The fact that the collection of all D-algebraic power series satisfies the Weierstrass preparation theorem was first proved in a more ad hoc way by van den Dries [6]. The notion of a tribe also shares some common properties with the notion of a Weierstrass system, as introduced by Denef and Lipshitz [3] and used in [6]. Our main theorem can be regarded as a simpler, effective and more systematic way to prove that certain types of power series form Weierstrass systems.

The idea behind our main algorithm is very simple: given a series of Weierstrass degree in , we just compute the solutions of the equation in inside a sufficiently large field of grid-based power series. This allows us to compute the polynomial which we know to be the Weierstrass polynomial associated to . Using the stability of the tribe under restricted monomial transformations, we will be able to compute as an element of .

The algorithms rely on our ability to compute with the auxiliary grid-based power series . For this reason, we briefly recall some basic facts about grid-based power series in section 3, as well as the basic techniques which are needed in order to compute with them.

2Common operations on power series

Let be a field of characteristic zero and denote

where we understand that is naturally included in for each . So each element is a power series in a finite number of variables.

We say that is effective if its elements can be represented by concrete data structures and if all field operations can be carried out by algorithms. We say that admits an effective zero test if we also have an algorithm which takes on input and which returns true if and false otherwise.

If is effective, then a power series is said to be computable if we have an effective bound for its dimension (so that ), together with an algorithm which takes on input and produces the coefficient of on output. We will denote the set of computable power series by .

Basic operations on power series

Let be a subset of . We will denote for each and say that is effective if . In this section, we will give definitions of several operations on power series and the corresponding closure properties that may satisfy. From now on, we will always assume that is at least a -algebra. It is also useful to assume that is inhabited in the sense that for all . For each , we will denote and . We say that is stable under differentiation if for all (whence ).

The above closure properties admit natural effective analogues. We say that is an effective -algebra if is an effective field, if the elements of can be represented by concrete data structures and the -algebra operations can be carried out by algorithms. We say that is effectively inhabited if there is an algorithm which takes on input and which computes . We say that is effectively stable under differentiation if there exists an algorithm which takes and on input and which computes .

Restricted division

We say that is stable under restricted division if whenever and are such that . If is effective, then we say that is effectively stable under restricted division if we also have an algorithm which computes as a function of , whenever . Here we do not assume the existence of a test whether (the behaviour of the algorithm being unspecified if ). More generally, given , we say that is stable under restricted division by if whenever , and that is effectively stable under restricted division by if this division can be carried out by algorithm.

Composition

Given , we let denote the evaluation of at . Given and with , we define the composition of and to be the unique power series with

We say that a power series domain is stable under composition if for any and with . If we also have an algorithm for the computation of , then we say that is effectively stable under composition.

We notice that stability under composition implies stability under permutations of the . In particular, it suffices that for to be inhabited. Stability under composition also implies stability under the projections with

If is also stable under restricted division by (whence under restricted division by any ), then this means that we may compute the coefficients of the power series expansion of with respect to by induction over :

Similarly, we obtain stability under the differentiation: for any and , we have

Implicit functions

Let with and . Assume that the matrix formed by the first columns of the scalar matrix

is invertible. Then the implicit function theorem implies that there exist unique power series , such that the completed vector with satisfies . We say that a power series domain satisfies the implicit function theorem (for implicit functions) if for the above solution of , whenever . We say that effectively satisfies the implicit function theorem if we also have an algorithm to compute as a function of .

We claim that satisfies the implicit function theorem for implicit functions as soon as satisfies the implicit function theorem for one implicit function and is stable under restricted division and composition. We prove this by induction over . For the statement is clear, so assume that . Since is invertible at least one of the must be non zero. Modulo a permutation of rows we may assume that . Applying the implicit function theorem to only, we obtain a function with . Differentiating this relation, we also obtain

for each . Setting , this yields in particular

Now consider the series . For each , we have

In particular,

By the induction hypothesis, we may thus compute series such that for all . Setting , we conclude that and

for all .

Restricted monomial transformations

Consider an invertible matrix with rational coefficients. Then the transformation

is called a monomial transformation, where is considered as a column vector. For a power series whose support satisfies , we may apply the monomial transformation to as well:

We say that is stable under restricted monomial transformations if for any and invertible matrix with , we have . We say that is effectively stable under restricted monomial transformations if we also have an algorithm to compute as a function of and . Notice that we do not require the existence of a test whether in this case (the behaviour of the algorithm being unspecified whenever ).

If has positive integer coefficients, then we always have and is trivially stable under the monomial transformation whenever is stable under composition.

Examples

We say that the -algebra with is a local community if is stable under composition, the resolution of implicit equations, and restricted division by . We say that is a tribe if is also stable under restricted monomial transformations. Effective local communities and tribes are defined similarly.

A power series is said to be algebraic if it satisfies a non trivial algebraic equation over the polynomial ring . Setting , this is the case if and only if the module is a -vector space of finite dimension. Using this criterion, it is not hard to prove that the set of algebraic power series is a tribe (and actually the smallest tribe for inclusion). Assume that is an effective field. Then an effective algebraic power series can be effectively represented as an effective power series together with an annihilator . It can be shown that is an effective tribe for this representation.

A power series is said to be D-algebraic if it satisfies a non trivial algebraic differential equation for each , where is a non zero polynomial in variables with coefficients in . We denote by the set of D-algebraic power series. If is an effective field, then effective D-algebraic power series may again be represented through an effective power series and differential annihilators of the above form. In [9], one may find more information on how to compute with D-algebraic power series, and a full proof of the fact that is an effective tribe (the proof being based on earlier techniques from [4, 5]).

3Grid-based series

Monomial monoids

In what follows, we will only consider commutative monoids. A monomial monoid is a multiplicative monoid with an asymptotic partial ordering which is compatible with the multiplication (i.e. and ). We denote by the set of infinitesimal elements in and by the set of bounded elements in . We say that has -powers if we also have a powering operation such that and for all and .

A monomial monoid is said to be effective if its elements can be represented by effective data structures and if we have algorithms for the multiplication and the asymptotic ordering . Since this implies the existence of an effective equality test. A monomial group is said to be effective if it is an effective monomial monoid with an algorithm for the group inverse. We say that is an effective monomial group with powers if we also have a computable powering operation.

Grid-based sets

A subset is said to be grid-based if there exist finite sets and such that

(1)

If is actually a group which is generated (as a group) by its infinitesimal elements, then we may always take .

If is an effective monomial monoid, then a grid-based subset is said to be effective if the predicate is computable and if finite sets and with (1) are explicitly given.

Grid-based series

Let be a field of characteristic zero. Given a formal series with , the set will be called the support of . We say that the formal series is grid-based if its support is grid-based and we denote by the set of such series. A grid-based series is said to be infinitesimal or bounded if resp. , and we denote by resp. the sets of such series.

In [8, Chapter 2] elementary properties of grid-based series are studied at length. We prove there that forms a ring in which all series with are invertible. In particular, if is a totally ordered group, then forms a field. Given a power series and grid-based series , there is also a natural definition for the composition .

Given a grid-based series the maximal elements of for are called dominant monomials for . If has a unique dominant monomial, then we say that is regular, we write for the dominant monomial of , and call the dominant coefficient of . If is totally ordered, then any non zero grid-based series in is regular.

Assume that and are effective. Then a grid-based series is said to be effective if its support is effective and if the map is computable. It can be shown that the set of computable grid-based series forms an effective -algebra.

Examples

Given an “infinitesimal” indeterminate , the set is a monomial monoid for the asymptotic ordering , and coincides with . Similarly, coincides with the field of Laurent series and with the field of Puiseux series in over . If is algebraically closed, then so is .

Given monomial monoids , one may form the product monomial monoid with for all . Then coincides with the set of power series , whereas coincides with the set of Laurent series .

Given monomial monoids , one may also form the set whose elements are ordered anti-lexicographically: if there exists an with and for all . The set should naturally be interpreted as (which it is isomorphic to ). The set is a field which contains , and this inclusion is strict if (notice also that ). If is algebraically closed, then is again an algebraically closed field (and again, we have ).

Cartesian representations

From now on, we will assume that is a monomial group which is generated as a group by its infinitesimal elements. Given a series , a Cartesian representation for is a Laurent series together with monomials such that . Given several series , and Cartesian representations for each of the , we say that these Cartesian representations are compatible if they are of the form for and . In [8, Proposition 3.12] we show that such compatible Cartesian representations always exist.

In [8, Chapter 3], we give constructive proofs of several basic facts about Cartesian representations and -based series to be introduced below. These constructive proofs can easily be transformed into algorithms, so we will only state the effective counterparts of the main results. First of all, in order to keep the number of variables in Cartesian representations as low as possible, we may use the following effective variant of [8, Lemma 3.13]:

Lemma 1.

Let be infinitesimal elements of an effective totally ordered monomial group with -powers, such that we have explicit expressions for as power products. Then we may effectively compute infinitesimal with .

-based power series

Let be a local community. We will say that is L-based if admits a Cartesian representation of the form with , and . The set of all such series forms a -algebra [8, Proposition 3.14]. If , and are effective, then any grid-based series in is computable. Moreover, we may effectively represent series in by Cartesian representations, and is an effective -algebra for this representation.

A Cartesian representation of is said to be faithful if for each dominant monomial of , there exists a dominant monomial of with . We have the following effective counterpart of [8, Proposition 3.19]:

Proposition 2.

Assume that , and are effective. Then there exists an algorithm which takes a series in on input and computes a faithful Cartesian representation with , and .

Faithful Cartesian representations are a useful technical tool for various computations. They occur for instance in the proof of the following effective counterpart of [8, Proposition 3.20]:

Proposition 3.

Assume that , and are effective. There exists an algorithm which takes an infinitesimal (or bounded, or regular) series on input and which computes a Cartesian representation such that is again infinitesimal (or bounded, or regular, respectively).

Solving power series equations

Assume now that is an effective field with an effective zero test and an algorithm for determining the roots in of polynomials in . Let be an effective local community over and an effective totally ordered monomial group. We notice that a grid-based series in can also be regarded as an ordinary power series in . We are interested in finding all infinitesimal solution of a power series equation

where . The Newton polygon method from [8, Chapter 3] can be generalized in a straightforward way to power series equations instead of polynomial equations and the effective counterpart of [8, Theorem 3.21] becomes:

Theorem 4.

There exists an algorithm which takes with on input and which computes all solutions of the equation with .

Given with , we may also consider as an element of . Let be the dominant of for this latter representation. The valuation of in is called the Weierstrass degree of . If is algebraically closed, then it can be shown that the number of solutions to the equation in Theorem 4 coincides with the Weierstrass degree, when counting with multiplicities.

4Effective Weierstrass preparation

Effective algebraic closures

Let be an effective field with an effective zero test. We may consider its algebraic closure as an effective field with an effective zero test, when computing non deterministically (we refer to [2] for more details about this technique, which is also called dynamic evaluation).

Let be an effective tribe over with an effective zero test. It is convenient to represent elements of by polynomials , where . The algebraic number is effectively represented using an annihilator and we may always take such that . It is a routine verification that forms again an effective tribe for this representation.

Consider a series , represented as , where is given by an annihilator of degree . Then we notice that we can compute a representation for as a element of . Indeed, whenever for some , then this means that there exists a monomial such that the coefficient of in is a polynomial of non zero degree in . On the other hand, , which means that we can compute an annihilator for of degree . Repeating this reduction a finite number of times, we thus reach the situation in which , so that .

Effective Weierstrass preparation

Let still be an effective tribe over with an effective zero test. Given , we recall that is said to have Weierstrass degree in if , but . In that case, the Weierstrass preparation theorem states that there exists unit and a monic polynomial of degree such that . The polynomial is called the Weierstrass polynomial associated to . We claim that and that there exists an algorithm to compute (and therefore the corresponding unit , since is effectively stable under restricted division):

Theorem 5. There exists an algorithm which takes a power series of Weierstrass degree on input and computes its Weierstrass polynomial as an element of .

Proof. Consider the effective totally ordered monomial group with -powers. We have a natural inclusion . Now consider . By theorem 4, we may compute all infinitesimal solutions to the equation in (we recall that there are such solutions, when counting with multiplicities, since is algebraically closed). Now consider

and let be the Weierstrass polynomial associated to . Since also admits the infinitesimal roots when considered as an element of , we have when considering as an element of . It follows that

Now consider a Cartesian representation for with . By Proposition 3, we may take to be infinitesimal. Since are infinitesimal and , Lemma 1 also shows that we may assume without loss of generality that . Completing the with additional elements if necessary, this means that we may compute an invertible matrix such that for all . In other words, with . Since and is effectively closed under restricted monomial transformations, we conclude that .

Effective Weierstrass division

Assume that has Weierstrass degree in and let . The Weierstrass division theorem states that there exists a quotient and a remainder in with

and . We claim that and once again belong to and that there exists an algorithm to compute them:

Theorem 6. There exists an algorithm which takes a power series of Weierstrass degree and on input and computes the quotient and remainder of the Weierstrass division of by as elements of .

Proof. Let be the distinct solutions of when considered as an equation in , and let be the multiplicity of each , so that . For each , we compute the polynomials

Using Chinese remaindering, we next compute the unique such that for each and . It is easily verified that coincides with the remainder of the Weierstrass division of by . In particular, and we may obtain as an element of in the same way as in the proof of Theorem 5. We obtain the quotient of the Weierstrass division by performing the restricted division of by .

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