Apparent Singularities of D-finite Systems

05/02/2017 ∙ by Shaoshi Chen, et al. ∙ 0

We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems. Ordinary points of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.

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1 Introduction

Ordinary linear differential equations allow easy access to the singularities of their solutions: every point which is a singularity of some solution of the differential equation must be a zero of the coefficient of the highest order derivative appearing in the equation, or a singularity of one of the other coefficients. For example,  is a solution of the equation , and the singularity at is reflected by the root of the polynomial  in front of the term in the equation. Unfortunately, the converse is not true: there may be roots of the leading coefficient which do not indicate solutions that are singular there. For example, all the solutions of the equation are constant multiples of , and none of these functions is singular at .

For a differential equation with polynomial coefficients and , the roots of are called the singularities of the equation. Those roots of such that the equation has no solution that is singular at  are called apparent. In other words, a root of is apparent if the differential equation admits linearly independent formal power series solutions in . Deciding whether a singularity is apparent is therefore the same as checking whether the equation admits a fundamental system of formal power series solutions at this point. This can be done by inspecting the so-called indicial polynomial of the equation at : if there exists a power series solution of the form , then  is a root of this polynomial.

When some singularity of an ODE is apparent, then it is always possible to construct a second ODE whose solution space contains all the solutions of the first ODE, and which does not have as a singularity. This process is called desingularization. The idea is easily explained. The key observation is that a point is a singularity if and only if the indicial polynomial at is different from or the ODE does not admit linearly independent formal power series solutions in . As the indicial polynomial at an apparent singularity has only nonnegative integer roots, we can bring it into the required form by adding a finite number of new factors. Adding a factor to the indicial polynomial amounts to adding a solution of the form to the solution space, and this is an easy thing to do using well-known arithmetic of differential operators. See [1, 4, 7, 12, 13] for an expanded version of this argument and [1, 2] for analogous algorithms for recurrence equations.

The purpose of the present paper is to generalize the two facts sketched above to the multivariate setting. Instead of an ODE, we consider systems of PDEs known as D-finite systems. For such systems, we define the notion of a singularity in terms of the polynomials appearing in them (Definition 3.1). We show in Theorem 3.4 that a point is a singularity of the system unless it admits a basis of power series solutions in which the starting terms are as small as possible with respect to some term order. Then a singularity is apparent if the system admits a full basis of power series solutions, the starting terms of which are not as small as possible. We then prove in Theorem 4.6 that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. The resulting system will be contained in the Weyl closure [23] of the original ideal, but unlike Tsai [23] we cannot guarantee that it is equal to the Weyl closure. Based on Theorem 3.4 and Theorem 4.6, we show how to remove a given apparent singularity (Algorithms 5.10 and 5.19), and how to detect whether a given point is an apparent singularity (Algorithm 5.13). At last, we present an algorithm for computing formal power series solutions of a D-finite system at apparent singularities.

2 Preliminaries

In this section, we recall some notions and conclusions concerning linear partial differential operators, Gröbner bases, formal power series, solution spaces and Wronskians for D-finite systems. We also specify notation to be used in the rest of this paper.

2.1 Rings of differential operators

Throughout the paper, we assume that is a field of characteristic zero and  is a positive integer. For instance, can be the field of complex numbers. Let be the ring of usual commutative polynomials over . The quotient field of is denoted by . Then we have the ring of differential operators with rational function coefficients , in which addition is coefficient-wise and multiplication is defined by associativity via the commutation rules

  • ;

  • ,

where is the usual derivative of  with respect to , . This ring is an Ore algebra [21, 9] and denoted by for brevity.

Another ring is , which is a subring of . We call it the ring of differential operators with polynomial coefficients or the Weyl algebra [22, Section 1.1].

A left ideal in is called D-finite if the quotient

is a finite dimensional vector space over

. The dimension of as a vector space over is called the rank of and denoted by .

For a subset of , the left ideal generated by is denoted by . For instance, let . Then  is D-finite because the quotient is a vector space of dimension  over . Thus, .

2.2 Gröbner bases

Gröbner bases in are well known [15] and implementations for them are available for example in the Maple package Mgfun [8] and in the Mathematica package HolonomicFunctions.m [18]. We briefly summarize some facts about Gröbner bases.

We denote by the commutative monoid generated by . An element of is called a term. For a vector , the symbol  stands for the term . The order of  is defined to be . For a nonzero operator , the order of  is defined to be the highest order of the terms that appear in  effectively.

Let  111In examples, we use the graded lexicographic ordering with . be a graded monomial ordering [10, Definition 1, page 55] on . Since there is a one-to-one correspondence between terms in and elements in , the ordering  on  induces an ordering on with if . Our main results on apparent singularities are based on the fact that there are at most finitely many terms lower than a given term. So we fix a graded ordering  on  in the rest of the paper.

For a nonzero element , the head term of , denoted by , is the highest term appearing in . The coefficient of  is called the head coefficient of and is denoted by . For a subset  of nonzero elements in , and stand for the sets of head terms and head coefficients of the elements in , respectively.

For a Gröbner basis in , a term is said to be parametric if it is not divisible by any term in . The set of exponents of all parametric terms is referred to as the set of parametric exponents of and denoted by . If is D-finite, then its rank is also called the rank of and denoted by , which is equal to .

2.3 Formal power series

Let be the ring of formal power series with respect to . For and , there is a natural action of on , which is denoted by . For , it is straightforward to verify that

(1)

For , the product is denoted by , and by . A formal power series can always be written in the form

where . Such a form is convenient for differentiation.

Taking the constant term  of a formal power series  gives rise to a ring homomorphism, which is denoted by . A direct calculation yields

(2)

Thus, we can determine whether a formal power series is zero by differentiating and taking constant terms, as stated in the next lemma.

Lemma 2.1.

Let . Then if and only if, for all ,

The following result appears in [11] for . But the proof applies literally also for arbitrary values of . Please see [25, Lemma 4.3.3] for a detailed verification.

Lemma 2.2.

Let and be polynomials in with

If has a power series expansion for each , then the constant term of is nonzero.

The fixed ordering on also induces an ordering on the monoid generated by in the following manner: if . A nonzero element can be written as

where  is nonzero. We call the initial exponent of .

2.4 Solutions and Wronskians

Basic facts about solutions of linear partial differential polynomials are presented in [16, Chapter IV, Section 5]. We recall them in terms of D-finite ideals. The first proposition is a special case of Proposition 2 in [16, page 152].

Proposition 2.3.

For a left ideal  with rank , there exists a differential field  containing  such that the solution space of  has dimension  over , where stands for the subfield of the constants in .

Such differential fields can also be constructed by the Picard-Vessiot approach (see, [20, Appendix D] or [6]). In the rest of this paper, we assume that is a differential field described in the above proposition. For a D-finite ideal , the solution space of  in  is denoted by . Likewise, for a finite-rank Gröbner basis , the solution space of  is simply denoted by .

The next proposition is an analog of differential Nullstellensatz for D-finite ideals. It is an easy consequence of Corollary 1 in [16, page 152].

Proposition 2.4.

Let  be a -dimensional linear subspace over . Then there exists a unique left ideal  of rank  such that

Furthermore, an operator belongs to  if and only if  annihilates every element of .

Linear dependence over constants can be determined by Wronskian-like determinants [16, Chapter II, Theorem 1], which implies that a finite number of elements in  are linearly independent over  if and only if they are linearly independent over any field of constants that contains .

Wronskian-like determinants are expressed by elements of via wedge notation in [19]. For and , the exterior product

is defined as a multi-linear function from to  that maps  to:

It follows from Theorem 1 in [16, Chapter II] that , …, are linearly independent over  if there exist  such that

Let  be a reduced and finite-rank Gröbner basis in the Wronskian operator of  is defined to be

The following proposition is Lemma 4 in [19] in slightly different notation.

Proposition 2.5.

Let , and .

  • The elements are linearly independent over  if and only if is nonzero.

  • Let  be the head term of an element  of , and let be linearly independent over . Set  and

    in which the elements of  are placed on the right-hand side of a product. Then

The two propositions listed above will be used to reconstruct a Gröbner basis from its solutions.

3 Ordinary points and singularities

Let be in the form where and …, are distinct. We say that is primitive if

For brevity, a Gröbner basis  in is said to be primitive if it is finite, reduced and its elements are primitive ones in . Every nontrivial left ideal in has a primitive Gröbner basis. The goal of this section is to characterize ordinary points of a primitive Gröbner basis of finite rank in terms of formal power series solutions.

3.1 Ordinary points

Our definitions of singularities and ordinary points are motivated by the material after [22, Lemma 1.4.21].

Definition 3.1.

Assume that is a primitive Gröbner basis of finite rank. A point is called an ordinary point of  if, for every , . Otherwise, it is called a singularity of .

The above definitions are compatible with those in the univariate case [1, 7]. Note that the origin is an ordinary point of if and only if each element of has a nonzero constant term.

Example 3.2.

Consider the Gröbner basis in

We find that and . So has no singularity.

Example 3.3.

Consider the Gröbner basis [19, Example 3] in

In this case, and and . The singularities of are

which are two lines in . In particular, the origin is a singularity.

3.2 Characterization of ordinary points

From now on, we focus on formal power series solutions of a primitive Gröbner basis around the origin, as a point in can always be translated to the origin, and we may assume that is algebraically closed when necessary.

Theorem 3.4.

Let be a primitive Gröbner basis of finite rank. Then the origin is an ordinary point of if and only if has many -linearly independent formal power series solutions whose initial exponents are exactly those in .

Proof.

Let , and , . Then we can write

Necessity. Assume that the origin is an ordinary point of . Then none of the ’s vanishes at the origin. We show how to construct formal power series solutions of by an approach described in [24].

We associate to each tuple an arbitrary constant . For a non-parametric term , let be its normal form with respect to . Although belongs to , there exists a power product of such that . Write

with . Set

(3)

Note that can be chosen to be any power product of such that  belongs to . Let

We claim that is a formal power series solution of , that is,

(4)

By (1) and Lemma 2.1, it suffices to prove

(5)

for all and . We proceed by Noetherian induction on the term order .

Starting with , we can write

(6)

where . It follows that

By (3),

which can be rewritten as

Since is a ring homomorphism, we have

We see that by (6).

Assume that is a term higher than and, for all with  and all ,

Reducing modulo , we have

where and are two power products of , and belongs to for all and . Moreover, belongs to . Applying the above equality to , we get

Applying to the above equality yields

By the induction hypothesis, for all with and for all . Thus,

Writing  with , we see that the above equality implies

It follows from (3) that

Since is nonzero, is equal to zero. This proves (5). Therefore, our claim (4) holds. Since there are many parametric terms, the D-finite system has many -linearly independent formal power series solutions with initial exponents in .

Sufficiency. Let . Assume that are -linearly independently formal power series solutions of whose the initial exponents are , …, , respectively. Without loss of generality, we assume further that, for all ,

where  stands for Kronecker’s symbol.

Let . By the above assumption, the constant term of is nonzero. So the formal power series is invertible in .

Let . By Proposition 2.5,

(7)

Since is primitive, we can write as

where and . By (7), we have

It follows from Lemma 2.2 that the constant term of is nonzero. Hence, the origin is an ordinary point of . ∎

The proof for the necessity of the above theorem also holds for an arbitrary left (not necessarily D-finite) ideal , provided that the origin is an ordinary point of . In addition, the above theorem also holds when the fixed ordering is not graded. But the results in the next section hinge on the assumption that is graded.

4 Apparent singularities

The goal of this section is to define apparent singularities of a primitive Gröbner basis of finite rank, and to characterize them.

Definition 4.1.

Let be a primitive Gröbner basis of rank .

  • Assume that the origin is a singularity of . We call the origin an apparent singularity of if has linearly independent formal power series solutions over .

  • Assume that is a primitive Gröbner basis of finite rank. We call  a left multiple of if

The above definition is compatible with the univariate case [1, Definition 5].

Example 4.2.

The solution space of the Gröbner basis

in is generated by . In this case,

Therefore, the origin is a singularity of . As has two -linearly independent formal power series solutions, the origin is an apparent singularity of .

Let be another Gröbner basis such that

We find that is a left multiple of  with rank .

Example 4.3.

The solution space of the Gröbner basis

in is generated by . In this case,

Therefore, the origin is an apparent singularity of .

Set

Let be another Gröbner basis with

We find that . By Definition 4.1 (ii), is a left multiple of  with .

For a subset  of , we denote by the set of initial exponents of nonzero elements in and call it the set of initial exponents of at the origin. Then is the dimension of , because any set of formal power series with distinct initial exponents are linearly independent over . For a primitive Gröbner basis , the origin is an ordinary point of  if and only if by Theorem 3.4, it is an apparent singularity if and only if  but by Definition 4.1.

Before characterizing apparent singularities, we prove two lemmas. The results in the first lemma are likely known, but we were not able to find proper references containing them.

Lemma 4.4.

Let and be -finite ideals in . Then

  • .

Proof.

Let  be a vector space over any field, and let and be two subspaces of . Set

and

It is straightforward to verify that the following sequence is exact222We thank Professor Yang Han for bring this exact sequence to our attention, which shortens our original proof..

It follows that is linearly isomorphic to In particular,

Setting , and , we prove the first assertion. The second assertion follows from the first one and Proposition 2.3.

For the last assertion, it is evident that . On the other hand,

Hence, . ∎

As a matter of notation, we define for . The second lemma illustrates a connection between parametric exponents and initial ones.

Lemma 4.5.

Let be a primitive Gröbner basis. Assume that has a basis in and for some . Then . Consequently, the origin is an ordinary point of .

Proof.

Assume that  form a basis of and their initial exponents are distinct. Then . Let . And set

Then  is a nonzero element in . For every  with , let , which belongs to . Then because is nonzero and the ordering  is graded. Since  annihilates , it vanishes on . It follows from Proposition 2.4 that belongs to the extended ideal , in which  is still a Gröbner basis. Thus, can be reduced to zero by . Accordingly, is not a parametric derivative of . In other words, is a subset of . Hence,  because and . The origin is an ordinary point by Theorem 3.4. ∎

Theorem 4.6.

Let  be a primitive Gröbner basis of rank . Assume that the origin is a singularity of . Then the origin is an apparent singularity of  if and only if it is an ordinary point of some left multiple of .

Proof.

Sufficiency. Assume that the origin is an apparent singularity of .

Set  For every , we denote by  the left ideal generated by  …, in . Note that the solution space of  is spanned by . Set

Then the two left ideals and have no solution in common except the trivial one because . It follows from Lemma 4.4 (iii) that

In particular, has dimension over , because and  have dimensions and , respectively. So . Let  be a primitive Gröbner basis of