.

This work has been supported by the ANR-10-BLAN 0109 LEDA project.

Computing with D-algebraic power series

by Joris van der Hoeven

CNRS, LIX, École polytechnique

91128 Palaiseau Cedex

France

Email: vdhoeven@lix.polytechnique.fr

January 4, 2015

Abstract

In this paper, we will present several algorithms for computing with D-algebraic power series. Such power series are specified by one or more algebraic differential equations and a sufficient number of initial conditions. The emphasis is not on the efficient computation of coefficients of such power series (various techniques are known for that), but rather on the ability to decide whether expressions involving D-algebraic power series are zero. We will both consider univariate and multivariate series and, besides the usual ring operations and differentiation, we will also consider composition, implicitly determined power series and monomial transformations.

Keywords: D-algebraic power series, algorithm, zero-test, implicit function

A.M.S. subject classification: 68W30, 34A09, 34A12

1Introduction

General introduction

Let be a field of characteristic zero. A power series is said to be D-algebraic if it satisfies a non trivial differential equation , where is a polynomial with coefficients in . The set of D-algebraic power series contains many classical transcendental functions, such as , , , etc., and it is closed under the ring operations, restricted division, differentiation and composition. This makes the differential ring of D-algebraic power series suitable as a framework for exact computations with mathematical expressions which involve transcendental functions.

The notion of D-algebraic power series admits a straightforward generalization to the multivariate context. In this case, we require the satisfaction of a non trivial algebraic differential equation with respect to each of the partial derivatives. The multivariate context allows for some additional operations, such as the resolution of implicit power series equations and general monomial transformations with rational powers. Again, the set of D-algebraic power series is stable under such operations.

There are two main aspects about 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 [2, 1, 9] or on relaxed power series evaluation [8, 6, 10].

On the other hand, there is the problem of deciding whether a given power series is zero. This problem is hard in the sense that we need to check the cancellation of an infinite number of coefficients. Therefore, a related question is how to represent power series in such a way that we can design such zero tests. We also notice the asymmetric aspect of the problem: given a non zero series , it is usually easy to prove that : it suffices to compute a non zero coefficient. However, if vanishes, then it is potentially difficult to establish a formal proof of this fact.

In this paper, we will focus on the second aspect. We will consider various representations for D-algebraic power series, show how to perform common operations on power series when using these representations, and also present several zero tests. All representations are based on a combination of differential equations satisfied by the power series and initial conditions. However, depending on additional properties of these equations, some representations are more suitable for performing common operations and zero testing.

For global computations with algebraic differential equations, it is convenient to use the classical framework of differential algebra [17, 14]. In addition, we need some technology in order to deal with initial conditions. One key ingredient is the determination of the number of initial conditions which are needed in order to guarantee that a power series solution of a system of differential equations is unique. For this, we will use a similar technique as the one introduced by Denef and Lipshitz in [4, 5], and develop this technique in further detail.

Structure of the paper and main results

Apart from a first section 2 with some reminders from differential algebra, the paper is subdivided into three main parts. In section 3, we first focus on the univariate case and the representation of a D-algebraic series by a single differential polynomial which annihilates together with a sufficient number of initial conditions. In section 4, we remain in the univariate setting, but switch to more flexible representations of D-algebraic series as solutions to systems of algebraic differential equations with sufficiently many initial conditions. In section 5, we generalize our results to the multivariate setting and also consider the additional operations of solving implicit equations, composition, and monomial transformations.

In section 3, we effectively represent D-algebraic power series by a pair , where is a computable power series (meaning that the function is computable) and a non zero differential polynomial such that . Specializing a more general result from [4, 5], we will show how to compute a number (called a root separation bound for at ) with the property that the equation admits no other solutions with . Moreover, if is a “non degenerate root” of (in the sense that , where is a simpler non zero differential polynomial, called the separant of ), then we actually obtain an explicit recurrence relation for in terms of for .

In section 3.3, we will exploit the existence of such a recurrence relation in the non degenerate case, by giving a first zero test for series in the differential field generated by . We will next strengthen the root separation bound by not only looking for other solutions of in , but also in . In section 3.4, this allows us to simplify the zero test (along similar lines as in [7]) and also widen its scope to power series which depend on a finite number of parameters (Remark 10). We finally consider the case when is ill specified as a degenerate root of . In sections 3.5 and 3.6, we give algorithms for computing root separation bounds in this case, as well as non degenerate annihilators.

In principle, annihilators of complex D-algebraic series (such as large expressions in other D-algebraic series) can be computed using brute force (Proposition 14). However, this technique is deemed to be very inefficient. For this reason, we introduce the more flexible framework of D-domains in section 4. In this framework, we express D-algebraic series as rational functions in a finite number of D-algebraic series which satisfy a special kind of system of algebraic differential equations with initial conditions. We will show how to adopt the major results from section 3 to this setting.

In section 5, we turn our attention to multivariate D-algebraic series. We will start by showing how to interpret a multivariate D-algebraic series in as a D-algebraic series in whose coefficients are D-algebraic series in . We next generalize the notion of a D-domain and show that the above reduction to the univariate case can be done at the level of D-domains. We conclude by giving some algorithms for some typical multivariate operations: the resolution of implicit equations, composition, and monomial transformations. In each of these cases, we will show how to avoid the computation of differential annihilators as much as possible, by remaining in the framework of multivariate D-domains.

Comparison with previous work

There are several approaches to the zero test problem for D-algebraic power series [16, 4, 5, 13, 18, 19, 15, 12] and we refer to [7] for a brief discussion. From a logical point of view, the most important decision problems for power series solutions to algebraic differential equations with initial conditions were settled in [4, 5]. One essential tool in this paper is the computation of a generalization of root separation bounds for more general decision problems. In a sense, this paper merely consists of specializations of these results to more specific problems. Nevertheless, we think that we introduced some noteworthy improvements which we will point out now.

It should first be emphasized that the papers [4, 5] are based on a more general decision procedure for testing whether systems of differential equations and inequations with initial conditions admit solutions. Efficiency is not the major concern here. The authors also do not attempt to state their results in terms of classical differential algebra, even though they are aware of this possibility. From our point of view, one main contribution of this paper is to isolate the part of the problem which can be dealt with classical differential algebra techniques from the part where initial conditions and root separation bounds come in (notice that [15] provides an interesting alternative way to achieve this goal). As a consequence, we can explicitly state our zero test algorithms, which we also believe to be more efficient.

Our approach also contains a few theoretical improvements. First of all, we mainly work over a so called “effective power series domain” instead of the constant field . For instance, we may take , where

is the differentially transcendental power series involved in the Euler-Maclaurin formula for the -function. Similarly, the conditions on the constant field are slightly weaker: we merely require an effective constant field with an algorithm for the computation of all positive integer roots of polynomials in . The improved zero test from section 3.4 also allows for power series which depend on parameters.

These theoretical improvements were actually introduced in [7], but the current presentation is a bit simpler and more systematic. In particular, we only need to consider logarithmic power series instead of logarithmic transseries in the correctness proof of the improved zero test from section 3.4. Furthermore, we added algorithms for the computation of non degenerate annihilators and root separation bounds in the degenerate case.

Section 4 contains no theoretical improvements, but we expect the more flexible framework of D-domains to be most suitable for practical computations. It is interesting to see that the root separation bounds and both zero tests from section 3 can be generalized to this setting. In particular, for the computation of root separation bounds, we introduced the dominant Hermite normal form, which seems interesting in its own right.

As we show in section 5, a multivariate D-algebraic series in can always be regarded as a D-algebraic series in whose coefficients are D-algebraic series in . From the logical point of view, decision problems for such series therefore reduce to their univariate counterparts. However, there are a few additional operations, such as solving implicit equations, extraction of coefficients and monomial transformations. Not only do we present algorithms for carrying out such operations, but we also discuss ways to make this efficient, in the framework of multivariate D-domains.

2Reminders from differential algebra

2.1Ritt reduction

Let us recall some standard notations from differential algebra. Let be a field of characteristic zero and let be a differential -algebra which is also an integral domain. We will mainly work with respect to a single derivation . Given a finite number of indeterminates , we will denote by or simply by the differential ring of differential polynomials in and by or its fraction field.

We will assume an admissible ranking on the set . For instance, we may take whenever or and . Given such a ranking, the leader of a differential polynomial is the highest variable occurring in when is considered as a polynomial in . We will denote by the leader of . Considering as a polynomial in , the leading coefficient is called the initial of , the separant, and we will denote . If has degree in , then the pair is called the Ritt rank of and such pairs are ordered lexicographically. We will also denote in that case.

Given , we say that is reducible with respect to if there exists an such that or and . The process of Ritt reduction provides us with a relation of the form

where , , and where is reduced with respect to . We will denote .

Given , we recall that stands for the differential ideal generated by . If , then we also denote and recall that

forms a differential ideal. We say that forms an autoreduced sequence if Ritt reduces to itself with respect to for each . In that case the set of differential polynomials which reduce to zero with respect to coincides with the differential ideal .

In section 5, we will also consider differential rings and differential polynomials with respect to a finite number of pairwise commuting derivations . In that case, has to be replaced with the set of all expressions with and . The notion of admissible rankings and Ritt reduction can be extended to this setting and we refer to classical textbooks on differential algebra [17, 14] for more details.

2.2Decompositions of differential polynomials

In order to explicitly write down a differential polynomial , it is convenient to use vector notation. We will index differential monomials by vectors where each is itself a finite sequence which may be padded with zeros whenever necessary. We denote

after which we may write

with . For a fixed degree , it will be convenient to denote by the homogeneous component of of degree :

The largest with will be called the degree of and we will denote it by . The smallest with will be called the differential valuation of and we denote it by . It will also be convenient to denote and .

2.3Additive conjugation

Given a differential polynomial and a “point” , it is often convenient to consider the additive conjugate of by , which is defined to be the unique differential polynomial with

for all . The coefficients of can be expressed directly by using a Taylor series expansion:

In particular, we get .

2.4Differential polynomials with power series coefficients

Assume now that and . Given , we will denote by its valuation in . This valuation naturally extends to differential polynomials in via the inclusion .

Assume now that and let be a homogeneous differential polynomial in a single variable of degree . Then we will denote by the polynomial with

For any series , we then have

We will also denote by the largest root of in , while taking if no such root exists.

2.5Logarithmic power series

For some purposes, we will occasionally consider logarithmic power series . Such series can still be considered as power series in and we will still denote by the valuation of in . The coefficients are polynomials in , and we will write . Notice that maps into itself for each .

Proposition 1. Consider a non-zero linear differential operator and write with and . Then there exists a unique operator with for every .

Proof. Let us first prove the existence of . If , then we may write with , and the equation admits the solution

since for every . For general , we may write with and take with

This proves the existence of . For any of degree in , we also notice that . This implies the uniqueness of .

3D-algebraic power series

3.1Univariate D-algebraic power series

Let be a field. Let be a differential -subalgebra of for with the property that for all and such that , we have . We will also call a power series domain.

A series is said to be D-algebraic over if it satisfies a non trivial differential equation with . In positive characteristic , any series satisfies the linear differential equation

whence any series is D-algebraic (indeed, annihilates all series in ). From now on, we will therefore assume that has characteristic zero.

Proposition 2. The series is D-algebraic if and only if admits finite transcendence degree over .

Proof. Assuming that is D-algebraic over , pick of minimal Ritt rank with . Then and is stable under the derivation , whence and . Conversely, assume that . Then satisfy a non trivial algebraic relation, whence is D-algebraic.

Proposition 3. The set of D-algebraic series over forms a power series domain.

Proof. Let . Then , whence and . Similarly, , whence and . Clearly, , so and . Assume finally that . Then , whence , so that .

Assume now that is an effective power series domain. The most obvious way to effectively represent a D-algebraic power series in is to represent it by a pair where is a computable series and a non trivial annihilator with . We define the multiplicity of as an annihilator of to be . We also say that the annihilator is non degenerate if , and notice that the multiplicity of a non degenerate annihilator is one. In order to make Proposition 3 effective, we will need a way to compute a non degenerate annihilator as a function of an arbitrary annihilator. This is not completely immediate, and we will postpone the presentation of an algorithm for doing so to the end of this section.

3.2Root separation bounds

Let be D-algebraic over with annihilator . Assume that there exists a number such that for any with and , we have . Then the smallest such number will be denoted by and we call it the root separation of at . It corresponds to the number of initial conditions which should be known in order to determine in a unique way as a root of . In fact always exists and, as we will see in the next section, we can give an algorithm to compute it under suitable assumptions.

Proposition 4. Assume that is D-algebraic over with annihilator . Then the following root separation bound holds:

(1)

Proof. Let and . Given with , we have

Now assume that . Then

whence

Since , we get , which entails .

The following proposition also provides us with a partial converse.

Proposition 5. Let and . Assume that and that , with . Then there exists a unique root of with .

Proof. Notice that implies that , so that is finite. Let . We have to show the existence of a unique series with and . We may decompose

Extracting the coefficient of in the relation now yields

For all , we have and only depends on . In other words, the relation (3) actually provides us with a recurrence relation for the computation of .

3.3A first effective zero test

We say that is effective if its elements can be represented effectively and if all field operations can be carried out by algorithms. We will call an effective diophantine field if all positive integer roots of polynomials over can be determined by algorithm. In particular, this means that admits an effective zero test, i.e. there exists an algorithm which takes an element of on input and which returns true if and false otherwise.

A power series is said to be computable, if there exists an algorithm for computing as a function of . The power series domain will said to be effective, if its elements are all effective power series and if the differential -algebra operations can be carried out by algorithms. We notice that the differential -algebra of all computable series is effective, although it does not admit an effective zero test.

Assume now that we are given an effective power series domain with an effective zero test over an effective diophantine field . Assume also that we are given an effective D-algebraic power series and an annihilator for . Assume finally that the annihilator has multiplicity one, so that we may compute and by expanding the power series coefficients of . In other words, the bound (1) from Proposition 4 provides us with an effective upper bound for .

Given a polynomial , we will now give an algorithm ZeroTest (or when we want to make the dependency on and explicit) for testing whether . In particular, this shows that the -algebra is again an effective power series domain.

Algorithm ZeroTest

Input:

Output: the result of the test

1

If then return the result of the test

2

If then return

3

If then return

4

If then return

5

Let

6

Return the result of the test

Proof. We first notice that recursive calls only occur for differential polynomials of a strictly smaller Ritt rank. This guarantees termination of the algorithm.

As to its correctness, we clearly have at line 2 whenever .

Assume now that we reach line 3 with . Then the degree of in its leader cannot be one, since this would imply , and we know that . Consequently, is a constant multiple of , whence for some and , and .

Assume now that we reach line 4. We have for some and . Since and , it follows that .

Assume finally that we reach step 5. If , then we clearly have , so assume that . Applying Proposition 5, we obtain a unique power series with and . It follows that , , and . The relation therefore implies . Applying Proposition 4 to , we obtain the bound . Since , we conclude that .

Remark 6. As a variant to the algorithm, we can also check whether for more than one at the same time. In that case, we keep reducing until we find a with and such that for all .

Remark 7. One drawback of the above zero test is that it does not apply to power series which depend on a finite number of parameters in . Indeed, this would require a root separation bound which is uniform in these parameters. Unfortunately, the largest integer root of a simple polynomial such as can become arbitrarily large, so the best uniform root separation bounds are usually .

3.4An improved zero test

In practical applications, the series is often the solution of a classical initial value problem, in which case . One disadvantage of the zero test from section 3.3 is that still depends on in quite an unpredictable way. In particular, even for simple , this quantity might a priori become arbitrarily large. In this section, we will give an improved version of our algorithm which does not have this drawback. The idea is to not only consider ordinary power solutions to our differential equations, but also logarithmic power series solutions in , as introduced in section 2.5.

Let be D-algebraic over with annihilator . Assume that there exists a number such that for any with , we have . Then the smallest such number will be denoted by and we call it the strong root separation of at . Proposition 4 naturally strengthens to this setting:

Proposition 8. Assume that is D-algebraic over with annihilator . Then the following strong root separation bound holds:

(4)

Proof. The proof is similar to the proof of Proposition 4 with the following change. Writing with , we now have

instead of (2), and where stands for a polynomial of degree at most in .

The consideration of logarithmic solutions leads to a better bound for the existence part of Proposition 5.

Proposition 9. Let and . Assume that and that , with . Then there exists a root of with .

Proof. The proof is analogous to the proof of Proposition 5, with the exception that (3) should be replaced by

For all , the right hand side again only depends on , but the constant term of the differential operator may vanish if . Yet, Proposition 1 still implies that the equation admits a solution in for any , which is sufficient for the existence of a solution to the equation .

In the proof of the algorithm ZeroTest, we only needed the existence of the solution with and . In view of what precedes, we may thus improve the algorithm as follows:

Algorithm ZeroTest

Input:

Output: the result of the test

1

If then return the result of the test

2

If then return

3

If then return

4

If then return

5

Let

6

Return the result of the test

Remark 10. Recall from Remark 7 that the zero test from the previous section does not work if or depends on a finite number of parameters in . One interesting aspect of the improved zero test is that we no longer require any root separation bounds which depend on , so the zero test still works if depends on parameters (when using the technique of dynamic evaluation [3] for examining the finite number of branches which can occur depending on algebraic conditions on the parameters). In fact, the original equation may also depend on parameters, as long as we have a uniform bound for .

3.5Effective root separation bounds

Assume that we are given an effective power series domain with an effective zero test over an effective diophantine field . Assume also that we are given an effective D-algebraic power series and an annihilator for . Let us show how the zero test algorithm from the previous section can be used in order to compute , thereby providing an effective bound for via Proposition 4.

Algorithm RootSeparationBound

Input: a computable D-algebraic series and with

Output: an upper bound for .

1

Let is an index with

2

Repeat the following

3

Let and

4

If then return

5

Let and be indices with , , , and set

6

Let

7

If then

8

Let be such that and

9

If for all with and , then return

10

Let be an index with

Proof. We first notice that strictly decreases at every iteration of the main loop, which implies the termination of our algorithm. As a loop invariant, we also notice that (whence ) whenever we are at line 3, which means that we indeed have an algorithm for the computation of . If , then the correctness of line 4 follows from Proposition 4. Otherwise, we construct such that and , whence the computability of at line 6. If at line 7, then Proposition 5 implies the existence and uniqueness of at line 8, and the relation (3) actually provides us with an algorithm to compute the coefficients of . Moreover and satisfy the assumptions for applying the algorithm to and its derivatives at line 9.

Now if , then in particular and by the uniqueness of . Consequently, the zerotests will indeed all succeed at line 9 and we will return a correct bound by Proposition 4. Conversely, if holds for all with and , then in particular . Since , we also have and , whence by Proposition 4 and . If , then this means that we will reach line 10 and find an index with and .

Remark 11. In practice, it is better to replace line 9 by a simultaneous zero test as outlined in Remark 6.

3.6Non degenerate annihilators

Proposition 12. There exists an algorithm which, given a computable D-algebraic series and with , computes an annihilator for of multiplicity one.

Proof. We may use a variant of the algorithm RootSeparationBound. Indeed, it suffices to return instead of in step 4, and instead of in step 9.

Proposition 13. There exists an algorithm which, given a computable D-algebraic series and with , computes a non degenerate annihilator for .

Proof. Using the previous proposition, we may assume without loss of generality that has multiplicity one. In particular, we have a zero test for elements in . Now let and keep replacing as long as . Then we will end up with a such that and .

We are now in a position to make Proposition 3 effective. We first need a general algorithm for computing algebraic dependencies.

Proposition 14. Let be an effective integral domain with an effective zero test. There exists an algorithm which takes polynomials on input, and which produces a relation with .

Proof. Let . Given , the set of power products with and contains at most elements. The degree of any polynomial in is bounded by , and the space of polynomials in of degree has rank as a free -module. Taking minimal such that , it follows that the set contains a non trivial -linear dependency , which we may compute using linear algebra.

If is a D-algebraic power series with non degenerate annihilator of order , then the -algebra is contained in the -algebra ,which is stable under .

Proposition 15. The set of D-algebraic series over forms an effective power series domain.

Proof. Let be represented by pairs and with . Applying Proposition 13, we may assume without loss of generality that the annihilators and are non degenerate, of orders and . Then

is an effective -algebra which is stable under . For , we may use Proposition 14 in order to compute an -algebraic relation between . Similarly, assuming that with , the algebra is stable under , so we may compute an -algebraic relation between .

Of course, the algorithm from the proof of Proposition 14 uses brute force for for finding algebraic relations, so any algorithm which relies on this method is deemed to be quite inefficient. In the section 4 below, we will discuss algorithms which avoid relying on Proposition 14 for the computation with D-algebraic series.

4D-domains

In the algorithms from the previous sections, we essentially represent D-algebraic power series by elements of , where is a computable root of a differential polynomial . Since is itself an effective power series domain with an effective zero test, we may also form towers and represent D-algebraic power series by elements of such towers. This generalization is useful for representing expressions involving , the -algebra operations, and other D-algebraic operations such as , , etc. Indeed, differential polynomials which annihilate such expression can quickly become quite large. In this section, we will introduce an even more convenient representation based on differential algebra, which generalizes the construction of towers and provides more flexibility for representing solutions to implicit equations at the end of section 5.5.

4.1Definition of a D-domain

An abstract D-domain over is a differential algebra over of finite transcendence degree over together with a differential -algebra morphism which we will call the evaluation mapping. A second abstract D-domain with evaluation mapping is said to be equivalent to if admits the same image as . Assuming that is an effective power series domain with an effective zero test over an effective diophantine field , we say that is effective if is computable and is computable for each ; in that case, we also say that is an effective D-domain.

A D-domain is an abstract D-domain of the form

where are such that

and where the leaders of are for certain . It will be convenient to lift the evaluation mapping to using . Then becomes simply the evaluation mapping at . In particular, is effective as soon as is a tuple of computable power series. We will call the D-domain unmixed if for each . We will call the D-domain Pfaffian if is of the form with for all .

Proposition 16.

  1. Given any D-domain over and , the series is D-algebraic over . If is effective, then we may effectively compute an annihilator for .

  2. Any D-algebraic series over is represented by an element of an unmixed D-domain over . If is effective and , then we may effectively compute such a .

Proof. Given a D-algebraic domain over and , the sequence contains non trivial algebraic dependencies which can be computed using Proposition 14. This proves a). Inversely, given , we may compute a non degenerate annihilator for using Proposition 13. Then with defines an unmixed D-domain in which represents .

Proposition 17. Any D-domain is equivalent to an unmixed D-domain. If is effective, then this reduction is effective.

Proof. Let be generators of the -algebra and let be the transcendence degree of over . For each , we may compute a non degenerate annihilator for using Proposition 13. Then with defines an unmixed D-domain which is equivalent to .

Proposition 18. Any D-domain is equivalent to a Pfaffian D-domain. If is effective, then this reduction is effective.

Proof. Let us first show that is equivalent to a D-domain with orders . Modulo the replacement of by , we may assume without loss of generality that and for all . Now consider formal variables with and . Let , and consider the -algebra morphism , with for and . Consider the set of all polynomials with if and . Let be the ranking on . We define a ranking on by setting whenever or and . Then the D-domain with is equivalent to .

Assuming that and for all , let us now show that is equivalent to a Pfaffian D-domain. We may assume that we ordered the variables such that . In particular, this implies that with . Let us prove by induction over that we may replace by a differential polynomial of the form with . So assume that the induction hypothesis is satisfied for all smaller . We have with . Let be the maximum of the degrees of and , and . Then substitution of for each with in and yields two polynomials and in such that . This means that we may replace by .

4.2Dominant Hermite normal forms

Before we generalize the zero test algorithms from section 3, we will need a way to asymptotically normalize systems of linear differential equations with power series coefficients. The normalization that we will use is an asymptotic variant of the Hermite normal form.

Let us first consider a square matrix . We say that is in Hermite normal form if is upper triangular and there exist integers such that

If has maximal rank , then we must have for each . It can be shown that there exists a unique unimodular matrix such that is in Hermite normal form. Moreover, and there is an algorithm for the computation of if is an effective field.

Let us now consider a matrix of rank . Recall that commutes with following the law . For each we will denote by the -th row of and by

its “skew dominant coefficient”, where

If , then we understand that . The matrix with rows will be called the row dominant matrix of . We say that is in dominant Hermite normal form if is in Hermite normal form. Any matrix with an inverse in and such that is in dominant Hermite normal form will be called a normalization matrix. We claim that such a matrix always exists. What is more, if the entries of are computable power series, then we may use the following algorithm to compute .

Algorithm NormalizationMatrix

Input: a matrix with computable coefficients and rank

Output: a normalization matrix in for

1

Let

2

Let be such that is in Hermite normal form

3

Let be such that

4

If has rank then return

5

Otherwise return

Proof. If , then is a normalization matrix of by construction. If , and assuming that is a normalization matrix of , then is a normalization matrix of , since is always invertible. This proves the correctness of the algorithm.

As to its termination, let and let be minimal such that the matrix with rows has rank for each . We claim that can only increase and that the -tuple can only decrease for the lexicographical ordering, once stabilizes.

Indeed, let , , and let be minimal such that the matrix with rows has rank for each . Then the rows are precisely the first rows of the Hermite normal form , and therefore form a matrix of rank . This shows that and for each .

Having proved our claims, we may assume without loss of generality that and have stabilized. Since the degrees of the polynomials entries of can only decrease, we may also assume that whence have stabilized. Furthermore, we may assume that our rows were ordered in such a way that for all . Modulo division of by powers of , we may finally assume that . Then and for all . This is repeatedly possible only if lies in the module spanned by for each . Since has rank , this means that we must have , which completes the termination proof.

4.3Root separation bounds for D-domains

Given an abstract D-domain , we claim that there exists a number such that for any alternative evaluation mapping on , we have whenever for all . The minimal such number will be called the root separation for and we denote it by . Our claim clearly holds when is an unmixed D-domain. Indeed, in this case, we have

with the notations from above. The general case reduces to this particular case by applying Proposition 17. However, since the computation of univariate differential polynomials which annihilate given elements of may be quite expensive, we would like to have a more direct bound. We first need a few preliminaries.

Consider linear differential polynomials . Any such polynomial can formally be viewed as an element of . Let be the matrix with . Assuming that this matrix has rank over , we may use the method from section 4.2 to compute a normalization matrix for . We will call such a a normalization matrix for . Applying to the column vector with entries we obtain a new column vector with “dominant reduced” entries . By construction, the dominant coefficient of is a polynomial of the form

with . We will denote by the polynomial with . We also denote by the largest root of in , or if no such root exists.

Now consider a D-domain with evaluation mapping . Let . Since , each is non zero and has the same leader as . In particular, the polynomials are linearly independent over . Consequently, there exists a normalization matrix for , whence and are well defined for all .

Proposition 19. Let be a D-domain with evaluation mapping and . Let be a normalization matrix for and . Then

Proof. Let for each . Given with , there exists a largest index with . We have

Assuming that , we also have

whence

Since , we get , which entails , and .

4.4An effective zero test for D-domains

In order to generalize the zero test from section 3.3 to the setting of D-domains, we first need a suitable counterpart of Proposition 5 in addition to Proposition 19.

Assume that we are given a differential ring where are such that for certain . Given and , we would like to solve the system of equations for with . Assuming that for each , we may define and as in the previous section. Since is a normalization matrix, there also exists a matrix with . Then we have and . Now consider

We have the following analogue of Proposition 5 for solving the equation for sufficiently small .

Proposition 20. With the above notations, if and , then there exists a unique with and .

Proof. By what precedes, it suffices to show that the equation admits a unique solution with . Let for each . Recall that we may write

for each , with . We now decompose each as

Extracting the coefficient of in the relation now yields

For all , we have and only depends on coefficients with and coefficients with . Hence (7) provides us with a recurrence relation for the computation of .

Algorithm ZeroTest

Input:

Output: the result of the test

1

If then return the result of the test

2

If then return

3

If then return

4

If for some then return

5

Let be an upper bound for

6

Let be such that for some

7

Let

8

Return the result of the test

Proof. The proof is analogous to the proof of the zero test algorithm from section 3.3.

4.5An improved zero test for D-domains

The zero test from the previous section may be further improved along similar lines as what we did in section 3.4. Given an abstract D-domain , the strong root separation for is the smallest number such that for any alternative “evaluation” mapping , we have whenever for all . The existence of such a number is shown in the same way as before and for D-domains we have the usual explicit bound:

Proposition 21. Let be a D-domain with evaluation mapping and . Let be a normalization matrix for and . Then

Proof. The proof is similar to the proof of Proposition 19. This time, , we again pick to be the largest index with , so that we may write with . In the same way as before, we now get

For , we again get , , and .

For the analogue of Proposition 20, we define

Proposition 22. With the above notations, if and , then there exists an with and .

Proof. The proof is similar to the proof of Proposition 20, except that (7) should be replaced by

where is the operator inverse of from Proposition 1.

In the algorithm ZeroTest from the previous section, it is now possible to replace by in the definition of .

5Multivariate D-algebraic series

5.1Multivariate power series domains and closure properties

Given a subring , we will denote by the intersection of with the fraction field of . We say that is a power series domain if is a differential ring with respect to the derivations such that and is stable under the substitutions .

A power series domain is said to be effective, if the differential -algebra operations can be carried out by algorithms and for each we have a computable mapping which takes on input and returns as a computable power series in . In particular, this means that every can be regarded as an computable power series in the sense that there exists an algorithm which takes on input and returns the coefficient of in on output. We also notice that the quotient field of an effective power series domain is an effective differential field, when representing fractions in the usual way. In particular, is an effective differential -algebra.

The above definitions can be generalized to countable dimension as follows. We let , where we regard each as being naturally included into . A subset is said to be a power series domain if is a power series domain for each . We say that is effective if each is and if we have an algorithm for computing an upper bound for the dimension of any given series .

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.

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 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 .

Consider an invertible matrix with rational coefficients. Then the transformation

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

A power series domain is said to be stable under monomial transformations if for any and invertible matrix with , we have . We say that is effectively stable under 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 ).

Given an effective power series domain which effectively satisfies each of the above closure properties (composition, implicit functions, and monomial transformations), it can be shown that an effective version of the Weierstrass preparation theorem holds. We refer to [11] for details.

5.2Multivariate D-algebraic power series

Let be a multivariate power series domain. Given a power series and , we may consider as a power series

in with coefficients in , and also as a power series in with coefficients in the fraction field of . If is D-algebraic over for this latter interpretation of , then we say that is D-algebraic in (or with respect to ). We say that is D-algebraic over if is D-algebraic in each of the variables .

Proposition 23. Given is D-algebraic over and , each coefficient of the power series expansion of in is D-algebraic over .

Proof. Given , let be a non trivial differential equation satisfied by in . Let denote the valuation of in . Modulo division of by , we may assume without loss of generality that . Now satisfies the non trivial equation . This shows that is D-algebraic over .

For , we will prove by induction that is D-algebraic over . So assume that are D-algebraic over , whence over . Given , the series

is D-algebraic in over , by Proposition 3. By what precedes, it follows that is D-algebraic in .

Proposition 24. The set of D-algebraic power series over forms a multivariate power series domain.

Proof. The stability of under the differential ring operations and division (when defined) directly follows from Proposition 3. The stability under the projections follows from Proposition 23.

From now on, denotes the set of differential polynomials with respect to the pairwise commuting derivations . Proposition 2 also admits a natural generalization:

Proposition 25. The series is D-algebraic if and only if admits finite transcendence degree over .

Proof. Assuming that is D-algebraic over , let be of minimal Ritt rank with , for each . Let . Then for each and is stable under the derivations . Consequently, and . Conversely, assume that . Then satisfy a non trivial algebraic relation for each , whence is D-algebraic.

Assume now that is an effective multivariate power series domain. We may effectively represent a D-algebraic series over by a tuple where is a computable power series in and an annihilator for with respect to , for each .

Proposition 26. Assume that is an effective multivariate power series domain over an effective diophantine field with an effective zero test. Then is an effective multivariate power series domain with an effective zero test. Moreover, for any , there exists an algorithm for computing the coefficients of the power series expansion of a given with respect to .

Proof. We prove the proposition by induction over . For , the result is trivial, so assume that and that we proved the result for all smaller .

Given , let us first show how to compute the coefficients of the power series expansion of a given with respect to . The induction hypothesis provides us with a zero test in . In the proof of Proposition 23, we thus have an algorithm for the computation of the valuation , and the remainder of this proof is constructive.

Let denote the quotient field of . We claim that is an effective diophantine field. Indeed, given a polynomial , an integer is a root of if and only if is a root of multiplied by the denominators of its coefficients. Without loss of generality, we may therefore assume that . After this reduction, is a root of if and only if is a root of the coefficient of in for all . Let be such that this coefficient is non zero and let be its largest root in (or if no such root exists). We may now check whether for and compute the largest root of in .

Any series in may be regarded as a univariate series in with coefficients in . By what precedes, is an effective diophantine field and we have an algorithm for the computation of the coefficients of . The zero tests from section 3 can therefore be used as zero tests for .

5.3Multivariate D-domains

Let be a multivariate power series domain. An abstract multivariate D-domain over is a differential -algebra for together with a differential -algebra morphism . A multivariate D-domain over is an abstract multivariate D-domain over of the form

where are such that

and where each only involves derivations of the form and admits a leader of the form for certain . We will denote by those elements of the fraction field of such that . We will also write for . We say that is effective if is an effective power series domain and is computable for each . We say that is unmixed if for all and . We say that is Pfaffian if is of the form with for all .

The proof of the following proposition is analogous to the proof of Proposition 16:

Proposition 27.

  1. Given any multivariate D-domain over and any , the series is D-algebraic over . If is effective, then we may effectively compute .

  2. Any multivariate D-algebraic series over is the element of a multivariate D-domain over . If is effective and , then we may effectively compute .

Corollary 28. Let be an effective multivariate D-domain over . Then there exists an algorithm for testing whether , for given .

Proof. This follows from Proposition 27 and the existence of a zero test in (Proposition 26).

The proofs of the following propositions are analogous to the proofs of Propositions 17 and 18:

Proposition 29.

Any multivariate D-domain is equivalent to an unmixed D-domain. If is effective, then this reduction is effective.

Proposition 30.

Any multivariate D-domain is equivalent to a Pfaffian D-domain. If is effective, then this reduction is effective.

Consider with and . Let be such that for all and assume that . Then we recall from differential algebra [17, 14] that the -polynomial of and is defined by

Given a D-domain as above, we say that is coherent if for all . Given an arbitrary effective D-domain , we may compute an equivalent effective coherent D-domain using the algorithm below, where we make use of the effective zero test from Corollary 28:

Algorithm MakeCoherent

Input: An effective multivariate D-domain

Output: A coherent effective multivariate D-domain which is equivalent to

1

Let

2

Repeat the following

3

Let

4

If there exists with , then set

5

Else if with and , then set

6

Else if with and , then set

7

Else if with and , then set

8

Else if with and , then set

9

Else return with the evaluation induced by

Proof. The main loop invariant states that only contains annihilators for the point , and each of the original differential polynomials Ritt reduces to zero with respect to . The termination follows from the fact that we only add new elements of smaller and smaller Ritt ranks to . At the end, the set is necessarily coherent and autoreduced, and such that . Since each original polynomial reduces to zero modulo , there must exist a corresponding polynomial with leader and . This shows that is indeed a D-domain.

5.4Extraction of coefficients and application to zero testing

By Proposition 26, there exists an algorithm for extracting the coefficients of multivariate D-algebraic power series with respect to a single variable . In the case of power series which are represented by elements of a D-domain , it would be nice to have a more efficient and systematic algorithm. In particular, given and , we would like to effectively construct a D-domain such that for any and any , we can produce a with . For this, it suffices that contains all coefficients of the form with and .

Theoretically speaking, we may construct using Proposition 26. As a first optimization, we claim that there exists a computable finite set such that we can take whenever . Indeed, for any , we may regard as a power series in with coefficients in the quotient field of . For sufficiently large , the coefficients of this power series are determined by a recurrence relation of type (3).

As a second optimization, assume that is Pfaffian and regular in , meaning that the valuations of the in all vanish. We will take

with for all , and where the differential ideal is constructed as follows. Given and , let

Since , extracting the coefficient of in the equation

yields a relation

for some . Now let be the set of all differential polynomials of the form . Then we may take . Notice that is again Pfaffian if is Pfaffian. However, regularity in the other variables is not necessarily preserved. Nevertheless, if for all , then the recursive extraction of coefficients only yields regular Pfaffian D-domains.

Remark 31. Given a regular D-domain in , it can be shown that the reduction to an equivalent Pfaffian D-domain used in the proof of Proposition 30 actually yields a regular Pfaffian D-domain in .

From what precedes, it follows in particular that there exists a constant such that we may take for all . Hence for any and any , we have . We may then reinterpret the D-domain as a univariate D-domain by only retaining the relations and using coefficients of the form with . This allows us to replace the theoretical zero test from Corollary 28 by one of the more efficient zero tests from section 4.

5.5An effective implicit function theorem

Let with and denote

Let be the submatrix spanned by the first columns of and the submatrix spanned by the last columns. Assume that is invertible. Then the implicit function theorem implies that there exist unique power series , such that the completed vector with satisfies

(9)

Forming the Jacobian matrix of , we let and denote the submatrices spanned by the first resp. last rows. Differentiating the relation (9), we obtain

Since , this yields

Given any , consider the row matrices and . Then

Let , and . Let and diagonal matrices with entries resp. . Denoting by the row vector of derivations with

the above relation implies

Notice that maps power series to row vectors of power series.

Assume now that for some effective multivariate D-domain of dimension over with

Assume also that the coordinate functions are among the . We will make the additional assumption that for all . In particular, setting , we have . We introduce a new evaluation mapping on by taking

We may formally extend into an evaluation from into . This evaluation is not compatible with the original derivations , but we do have

for all , and for each of the derivations . Since is finite dimensional as a -algebra, Proposition 14 implies that satisfy a computable non trivial -algebraic relation for each and . Using Proposition 13, we may assume without loss of generality that these relations are non degenerate. We now define a new multivariate D-domain for the derivations , by taking

and taking the evaluation mapping as in (10). By construction, for each , so is D-algebraic.

The additional assumption that for all is quite benign. Indeed, for any index with , postcomposition of with means in particular sending to . This can also be achieved by extracting the coefficient of in . Consequently, modulo a finite number of extraction of coefficients, we may assume without loss of generality that for all . We have proved:

Proposition 32. The set of D-algebraic power series over an effective diophantine field is effectively satisfies the implicit function theorem.

Proposition 33. The set of D-algebraic power series over an effective diophantine field is effectively stable under composition.

Proof. Given and with , consider the power series . The above algorithm allows us to compute with and , as well as the result of the composition (when considering as an element of ), which coincides with .

The construction of is somewhat unsatisfactory since the computation of individual -algebraic relations using Proposition 14 is usually quite expensive. In the frequent situation that for all , we may use the following more efficient technique. Using Proposition 30, we may assume without loss of generality that the original relations are of the form with . By assumption, we have for all . With the above notations, let denote the horizontal concatenation of the matrices and , so that for all and has coefficients in . Modulo the relations , we may then write

For each and , this means that there exists a power product of the and a polynomial with

This allows us to take in the construction of the multivariate D-domain instead of the relation found using Proposition 14.

5.6Effective monomial transformations

Proposition 34. Let be an effective power series domain over an effective diophantine field . Then is effectively stable under monomial transformations.

Proof. Let be an effective D-domain over , let and . Given an invertible matrix with , we have to show how to compute . Given a row vector , we will denote . Then for any power series and any , we notice that

Since , the elements satisfy an -algebraic dependency which can be computed using Proposition 14. Applying this result for all of the form , we obtain D-algebraic relations over for in the individual derivations .

The above proposition could in principle be used for computing with monomial transforms of elements in an effective D-domain . However, in a similar way as it is more efficient to keep fractions in their symbolic forms when computing with fractions in , it is often more efficient to keep monomial transforms in their symbolic form when computing with them.

More precisely, we first notice that monomial transformations with are somewhat easier, since they can be computed using our algorithms for composition. Now two monomial transformations and are said to be compatible if there exist invertible matrices with . Given with and , let , and . Since and have coefficients in , we may construct a D-domain with . Now and . In particular, any can be represented by for some .

From the geometrical point of view, the notion of compatibility can be interpreted as follows. Let be an open subset of and . Let be the subset of of series with . Given a monomial transformation , we call its associated cone. Given two monomial transformations and with associated cones and , it is not hard to check that these transformations are compatible if and only if .

5.7Heuristic zero testing

It should be emphasized that zero testing of power series is a quite asymmetric problem in the sense that it is usually easy to prove that a non zero series is indeed non zero (it suffices to find a non zero coefficient), whereas it is often hard to prove vanishing series to be zero. Since exact zero tests are so slow, it is often preferable to use heuristic zero tests instead. In fact, heuristic zero tests can be combined with genuine zero tests: for a complex computation which involves many zero tests, we first perform all zero tests heuristically. At the end of the computation, we collect all series which were heuristically assumed to be zero in order to compute the result, and we apply an exact zero test to these series.

The most obvious heuristic zero test for a univariate D-algebraic series is to compute all coefficients up to a fixed order. Even for large orders, these coefficients can be computed efficiently using Newton's method or relaxed power series evaluation [2, 8, 1, 9, 6, 10]. In the case of multivariate D-algebraic series , one simple idea is to generate a random scalar vector and to test whether . The composition can be computed efficiently using the algorithm(s) from section 5.5. Here we notice the remarkable fact that the derivation

for which

does not depend on . This makes it actually possible to design another exact zero test for multivariate D-algebraic series: compute a univariate D-domain capable of representing where is treated as a formal parameter, and test whether .

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