Additive Decompositions in Primitive Extensions

02/07/2018 ∙ by Shaoshi Chen, et al. ∙ 0

This paper extends the classical Ostrogradsky-Hermite reduction for rational functions to more general functions in primitive extensions of certain types. For an element f in such an extension K, the extended reduction decomposes f as the sum of a derivative in K and another element r such that f has an antiderivative in K if and only if r=0; and f has an elementary antiderivative over K if and only if r is a linear combination of logarithmic derivatives over the constants when K is a logarithmic extension. Moreover, r is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D-finite.

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1 Introduction

Symbolic integration, together with its discrete counterpart symbolic summation, nowadays has played a crucial role in building the infrastructure for applying computer algebra tools to solve problems in combinatorics and mathematical physics [15, 16, 27]. The early history of symbolic integration starts from the first tries of developing programs in LISP to evaluate integrals in freshman calculus symbolically in the 1960s. Two representative packages at the time were Slagle’s SAINT [28] and Moses’s SIN [19]

which were both based on integral transformation rules and pattern recognition. The algebraic approach for symbolic integration is initialized by Ritt 

[25] in terms of differential algebra [14], which eventually leads to the Risch algorithm for the integration of elementary functions [23, 24]. The efficiency of the Risch algorithm is further improved by Rothstein [26], Davenport [11], Trager [29], Bronstein [6, 7] etc. Some standard references on this topic are Bronstein’s book [8] and Raab’s survey [22] that gives an overview of the Risch algorithm and its recent developments.

The central problem in symbolic integration is whether the integral of a given function can be written in “closed form”. Its algebraic formulation is given in terms of differential fields and their extensions [14, 8]. A differential field  is a field together with a derivation that is an additive map on satisfying the product rule for all . A given element  in is said to be integrable in if for some . The problem of deciding whether a given element is integrable or not in  is called the integrability problem in . For example, if is the field of rational functions, then for we can find , while for no suitable exists in . When is not integrable in , there are several other questions we may ask. One possibility is to ask whether there is a pair  in  such that , where  is minimal in some sense and if  is integrable. This problem is called the decomposition problem in . Extensive work has been done to solve the integrability and decomposition problems in differential fields of various kinds.

Abel and Liouville pioneered the early work on the integrability problem in the 19th century [25]. In 1833, Liouville provided a first decision procedure for solving the integrability problem on algebraic functions [18]. For other classes of functions, complete algorithms for solving the integrability problem are much more recent: 1) the Risch algorithm [23, 24] in the case of elementary functions was presented in 1969; 2) the Almkvist–Zeilberger algorithm [2] (also known as the differential Gosper algorithm) in the case of hyperexponential functions was given in 1990; 3) Abramov and van Hoeij’s algorithm [1] generalized the previous algorithm to the general -finite functions of arbitrary order in 1997.

The decomposition problem was first considered by Ostrogradsky [20] in 1845 and later by Hermite [13] for rational functions. The idea of Ostrogradsky and Hermite is crucial for algorithmic treatments of the problem, since it avoids the root-finding of polynomials and only uses the extended Euclidean algorithm and squarefree factorization to obtain the additive decomposition of a rational function. This reduction is a basic tool for the integration of rational functions and also plays an important role in the base case of our work. We will refer this reduction as to the rational reduction in this paper. The rational reduction has been extended to more general classes of functions including algebraic functions [29, 9], hyperexponential functions [12, 4], multivariate rational functions [5, 17], and more recently including -finite functions [10, 30]. Blending reductions with creative telescoping [2, 31] leads to the fourth and most recent generation of creative telescoping algorithms, which are called reduction-based algorithms [3, 4, 5, 9, 10].

The telescoping problem can also be formulated for elementary functions. Two related problems are how to decide the existence of telescopers for elementary functions and how to compute one if telescopers exist. Reduction algorithms have been shown to be crucial for solving these two problems. This naturally motivates us to design reduction algorithms for elementary functions.

In this paper, we extend the rational reduction to elements in straight and flat towers of primitive extensions (see Definition 3.5). Our extended reductions solve the decomposition problems in such towers without solving any Risch equations (Theorems 4.8 and 5.15), and determine elementary integrability in such towers when primitive extensions are logarithmic (Theorem 6.1).

The remainder of this paper is organized as follows. We present basic notions and terminologies on differential fields, and collect some useful facts about integrability in primitive extensions in Section 2. We define the notions of straight and flat towers, and describe some straightforward reduction processes in Section 3. Additive decompositions in straight and flat towers are given in Sections 4 and 5, respectively. The two decompositions are used to determine elementary integrability in Section 6. Examples are given in Section 7 to illustrate that the decompositions may be useful to study the telescoping problem for elementary functions that are not -finite.

2 Preliminaries

Let be a differential field of characteristic zero, and let denote the subfield of constants in . Let be a differential field extension of . An element of is said to be primitive over if  belongs to . If is primitive and transcendental over with , then it is called a primitive monomial over , which is a special instance of Liouvillian monomials [8, Definition 5.1.2].

Let be a primitive monomial over in the rest of this section. An element is said to be proper with respect to or -proper for brevity if the degree of its numerator in is lower than that of its denominator. In particular, zero is -proper. It is well-known that can be uniquely written as the sum of a -proper element and a polynomial in . They are called the fractional and polynomial parts of , and denoted by and , respectively.

Let be a polynomial in . The degree and leading coefficient of are denoted by and , respectively. By [8, Thereom 5.1.1], is squarefree if and only if . A -proper element is -simple if its denominator is squarefree. Note that -simple elements are not necessarily -proper in [8], but they are assumed to be -proper in this paper without loss of generality.

For , we use to denote the set . If is a -linear subspace, so is . For , let

In particular, .

For , Algorithm HermiteReduce in [8, page 139] computes a -simple element and a polynomial such that This algorithm is an extension of the rational reduction by Ostrogradsky and Hermite. For rational functions, we have since all polynomials have polynomial antiderivatives. Algorithm HermiteReduce is fundamental for our approach to additive decompositions in primitive extensions.

Lemma 2.1.

Let be a -simple element in . Then if 

Proof.

Suppose that . Since is -proper, there exists a nontrivial irreducible polynomial dividing the denominator of . Since there exist and  such that The order of at is equal to . But the order of at is either nonnegative or less than by Theorem 4.4.2 (i) in [8], and the order of  at is nonnegative, a contradiction.  

Every element is congruent to a unique -simple element modulo  by Algorithm HermiteReduce and Lemma 2.1. We call the Hermitian part of with respect to , denoted by . The map is -linear on . Its kernel is equal to . Thus, two elements have the same Hermitian parts if they are congruent modulo . This observation is frequently used in the sequel.

Now, we collect some basic facts about primitive monomials. They are either straightforward or scattered in [8]. We list them below for the reader’s convenience.

Lemma 2.2.

If and , then there exists such that

Proof.

Assume  for some . Then by Theorem 4.4.2 (i) in [8]. Set and . Then by Lemma 5.1.2 in [8]. Assume that for some . Then

Since , we have that and . Hence, with .  

The next lemma will be used to decrease the degree of a polynomial modulo . Its proof is a straightforward application of integration by parts.

Lemma 2.3.

We have for all  and .

Recall that an element in is said to be a logarithmic derivative in if for some nonzero element .

Lemma 2.4.

If  is a -linear combination of logarithmic derivatives in , then where is a -linear combination of logarithmic derivatives in .

Proof.

It suffices to assume that is a logarithmic derivative in , because the map is -linear.

If , then we choose , which equals . Otherwise, there exist two monic polynomials and such that by the logarithmic derivative identity in [8, page 104]. Note that is -simple by Lemma 5.1.2 in [8] and is in . Thus, and .  

3 Primitive extensions

Let be a differential field of characteristic zero. Set . Consider a tower of differential fields

(3.1)

where for all with . The tower given in (3.1) is said to be primitive over if is a primitive monomial over for all with . The notation introduced in (3.1) will be used in the rest of the paper.

Remark 3.1.

The derivatives are linearly independent over , since in (3.1).

The following lemma tells us a way to modify the leading coefficient of a polynomial in via integration by parts and Algorithm HermiteReduce.

Lemma 3.2.

Let the tower (3.1) be primitive with . Then, for all and , there exist a -simple element and a polynomial such that

Proof.

By Algorithm HermiteReduce, there are with being -simple, and such that . Then . Applying Lemma 2.3 to the term , we see that the lemma holds.  

Let be the purely lexicographic ordering on the set of monomials in with . For all  with and with , the head monomial of , denoted by , is defined to be the highest monomial in appearing in with respect to . The head coefficient of , denoted by , is defined to be the coefficient of , which belongs to . The head coefficient of zero is set to be zero. The monomial ordering induces a partial ordering on , which is also denoted by .

Example 3.3.

Let . Viewing as an element of , we have and , while, viewing as an element of , we have and ,

The next lemma will be used in Section 5. We present it below because it holds for primitive towers.

Lemma 3.4.

Let . For a polynomial , there are polynomials such that , and that is -simple for all with . Moreover, for all with .

Proof.

We proceed by induction on . If , then set , because there is no requirement on . Assume that and that the lemma holds for .

Let and . By Lemma 3.2,

where is -simple and . Then there exist such that for some in and is -simple for all when by the induction hypothesis. Furthermore, we set . By Lemma 2.3,

(3.2)

We need to argue inductively on . If , then it is sufficient to set for all with , as Assume that and that the lemma holds for all polynomials in . By (3.2) and the induction hypothesis on , we have

where is in , is -simple when , and . Set . Then . Since is if and is if , the requirements on each with is fulfilled. The induction on is completed, and so is the induction on .  

Definition 3.5.

The tower given in (3.1) is said to be straight if  for all with . The tower is said to be flat if  for all with .

Example 3.6.

Let with the usual derivation in . Let

Then the tower

is straight, while the tower

is flat. They contain no new constants by Theorem 5.1.1 in [8].

In this paper, we consider additive decompositions for elements in either straight or flat towers with .

Lemma 3.7.

Let the tower (3.1) be primitive. Assume that  is equal to with the usual derivation in . Then is nonzero. Moreover, is nonzero for all with if (3.1) is flat.

Proof.

By the rational reduction, for some with being -simple. Then is nonzero, since . The second assertion can be proved similarly.  

Example 3.8.

Let , with the usual derivation in , and each be logarithmic in (3.1) for all  with . By Lemma 2.4, , where for all with . The tower is straight if and only if for all  with . In addition, as .

4 Straight towers

In this section, we assume that the tower (3.1) is straight and that with the usual derivation with respect to . The subfield of constants is denoted by in recursive definitions and induction proofs to be carried out.

Our idea is reducing a polynomial in to another of lower degree via integration by parts, whenever it is possible. The notion of -rigid elements describes such that cannot be congruent to a polynomial of degree lower than  modulo .

Definition 4.1.

An element is said to be -rigid if . Let with

We say that is -rigid if is -simple, for any nonzero , and is -rigid.

Note that zero is -rigid, because is nonzero.

Example 4.2.

Let , and . Let

Then is -rigid if and is -rigid if .

The next lemma, together with Lemma 2.2, reveals that a nonzero polynomial in with a -rigid leading coefficient has no antiderivative in .

Lemma 4.3.

Let be -rigid. If

(4.1)

for some , then both and are zero.

Proof.

We proceed by induction on . If , then by Definition 4.1. Thus, . Consequently, because and .

Assume that and that the lemma holds for . Set . Then , since is -simple by Definition 4.1. Applying the map to (4.1), we have by Lemma 2.1. Hence, and by Definition 4.1.

Set . Then (4.1) becomes , which, together with Lemma 2.2, implies that for some . It follows from the induction hypothesis that is zero, and so is . Thus, is zero.  

In , we define a class of polynomials that have no antiderivatives in .

Definition 4.4.

For , a polynomial in is said to be -straight if its leading coefficient is -rigid.

Proposition 4.5.

Let be a -straight polynomial. Then if .

Proof.

If , then by Definition 4.1. Otherwise, for some by Lemma 2.2. Then by Lemma 4.3. Consequently, .  

Next, we reduce a polynomial to a -straight one.

Lemma 4.6.

For , there exists a -straight polynomial with such that

Proof.

If , then we choose . Assume that is nonzero. We proceed by induction on .

If , then , as every element of  has an antiderivative in the same ring.

Assume that and that the lemma holds for . Let with degree and leading coefficient . By Algorithm HermiteReduce, there are with being -simple, and such that

We are going to concoct a new expression for such that

(4.2)

where , and is -rigid. The expression helps us decrease degrees. To this end, we consider two cases.

Case 1. Assume for any . By the induction hypothesis, there exists a -straight polynomial such that for some . Then , where and .

Case 2. Assume for some . By the induction hypothesis, there exists a -straight polynomial such that for some . Then , where and .

In both cases, is -rigid by Definition 4.1.

If , then . By (4.2), we have . Let , which is -straight by Definition 4.4.

Assume that and each polynomial in  is congruent to a -straight polynomial modulo . By (4.2), Lemma 2.3 and the equality , we have

for some . If , then set Otherwise, applying the induction hypothesis on to yields a -straight polynomial with .  

Example 4.7.

Consider the integral

With the notation introduced in Example 4.2, we reduce the integrand . We have that . Since is not -rigid, can be reduced. In fact, . By Lemma 2.3 and a straightforward calculation, we get

Since is -rigid, we have that is -straight. Hence, has no antiderivative in by Proposition 4.5.

Below is an additive decomposition in a straight tower.

Theorem 4.8.

For , the following assertions hold.

  1. There exist a -simple element and a -straight polynomial such that

    (4.3)
  2. if and only if both and in (4.3) are zero.

  3. If , where is a -simple element and , then and

Proof.

(i) By Algorithm HermiteReduce, there exist a -simple element and a polynomial such that

By Lemma 4.6, can be replaced by a -straight polynomial .

(ii) Since , the congruence (4.3) becomes . Applying the map to the new congruence, we have , because . Thus, by Proposition 4.5.

(iii) Since , we have by Lemma 2.1. If , then is -straight, because equals . So by Proposition 4.5, a contradiction.  

Example 4.9.

Consider the integral

The integrand is , in which the notation is introduced in Example 4.2. By Algorithm HermiteReduce, we have

By Theorem 4.8 and Example 4.7, has no antiderivative in .

5 Flat towers

In this section, we let the tower (3.1) be flat. The ground field will be specialized to later in this section. We are not able to fully carry out the same idea in Section 4, because for all . This spoils Lemma 4.3 and Proposition 4.5. So we need to study integrability in a flat tower differently.

This section is divided into two parts. First, we extend Lemma 2.3 to the differential ring . Second, we present a flat counterpart of the results in Section 4.

5.1 Scales

Let us denote by . For a monomial in , …, , the -linear subspace is denoted by . The notion of scales is motivated by the following example.

Example 5.1.

Let , and and . And let . Using integration by parts, we find three congruences

The first and third congruences lead to monomials lower than  and , respectively. But the second one leads to , which is higher than . The notion of scales aims to prevent the second congruence from the reduction to be carried out.

Definition 5.2.

Let and The scale of with respect to is defined to be if , …, and . Let . The scale of with respect to is defined to be . The scale of with respect to is denoted by .

Example 5.3.

Let , and . Regarding and as elements in , we have that and ; while, regarding them as elements in , we have that and .

Notably, if , then the scale of with respect to is equal to , which varies as does. Otherwise, the scale is fixed by no matter in which ring lives.

The next lemma extends Lemma 2.3 and indicates what kind of integration by parts will be used for reduction.

Lemma 5.4.

Let be a monomial in …, and . Then the followings hold.

  • Let . Then, for all ,

Proof.

(i) It follows from integration by parts and the fact that   belongs to .

(ii) Set and for , …, .

If , then and The assertion clearly holds. Assume that with . Then Note that belongs to by a direct use of integration by parts. Set . Then the term  is equal to . Integration by parts leads to

(5.1)

Then if for all with . So belongs to by (5.1). Otherwise, for some with . Then each monomial in is of total degree and is of degree in . So . Consequently, by (5.1).  

In the rest of this section, we let with the usual derivation in . By Lemma 3.7, we may further assume that is nonzero and -simple for all with .

Definition 5.5.

For every with , an element of  is said to be -rigid if either it is equal to zero or it is -simple and is not a -linearly combination of .

Proposition 5.6.

For , there exists such that

and that is -rigid, where . Moreover, .

Proof.

Set if . Assume and . By the rational reduction, for some with