Apparent Singularities of D-finite Systems

1 Introduction

Ordinary linear differential equations allow easy access to the singularities of their solutions: every point $α$ which is a singularity of some solution $f$ 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, $x^{- 1}$ is a solution of the equation $x f^{'} (x) + f (x) = 0$ , and the singularity at $0$ is reflected by the root of the polynomial $x$ in front of the term $f^{'} (x)$ 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 $x f^{'} (x) - 5 f (x) = 0$ are constant multiples of $x^{5}$ , and none of these functions is singular at $0$ .

For a differential equation $p_{0} (x) f (x) + \dots + p_{r} (x) f^{(r)} (x) = 0$ with polynomial coefficients $p_{0}, \dots, p_{r}$ and $p_{r} \neq 0$ , the roots of $p_{r}$ are called the singularities of the equation. Those roots $α$ of $p_{r}$ such that the equation has no solution that is singular at $α$ are called apparent. In other words, a root $α$ of $p_{r}$ is apparent if the differential equation admits $r$ linearly independent formal power series solutions in $x - α$ . 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 $α$ : if there exists a power series solution of the form $(x - α)^{ℓ} + \dots$ , then $ℓ$ is a root of this polynomial.

When some singularity $α$ 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 $α$ as a singularity. This process is called desingularization. The idea is easily explained. The key observation is that a point $α$ is a singularity if and only if the indicial polynomial at $α$ is different from $n (n - 1) \dots (n - r + 1)$ or the ODE does not admit $r$ linearly independent formal power series solutions in $x - α$ . 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 $n - s$ to the indicial polynomial amounts to adding a solution of the form $(x - α)^{s} + \dots$ 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 $K$ is a field of characteristic zero and $n$ is a positive integer. For instance, $K$ can be the field of complex numbers. Let $K [x] = K [x_{1}, \dots, x_{n}]$ be the ring of usual commutative polynomials over $K$ . The quotient field of $K [x]$ is denoted by $K (x)$ . Then we have the ring of differential operators with rational function coefficients $K (x) [\partial_{1}, \dots, \partial_{n}]$ , in which addition is coefficient-wise and multiplication is defined by associativity via the commutation rules

$\partial_{i} \partial_{j} = \partial_{j} \partial_{i}$ ;
$\partial_{i} f = f \partial_{i} + \frac{\partial f}{\partial x_{i}} for % each f \in K (x)$ ,

where $\frac{\partial f}{\partial x_{i}}$ is the usual derivative of $f$ with respect to $x_{i}$ , $1 \leq i, j \leq n$ . This ring is an Ore algebra [21, 9] and denoted by $K (x) [\partial]$ for brevity.

Another ring is $K [x] [\partial] := K [x_{1}, \dots, x_{n}] [\partial_{1}, \dots, \partial_{n}]$ , which is a subring of $K (x) [\partial]$ . We call it the ring of differential operators with polynomial coefficients or the Weyl algebra [22, Section 1.1].

A left ideal $I$ in $K (x) [\partial]$ is called D-finite if the quotient $K (x) [\partial] / I$

is a finite dimensional vector space over

K (x)

. The dimension of

K (x) [\partial] / I

as a vector space over

K (x)

is called the rank of

I

and denoted by

rank (I)

For a subset $S$ of $K (x) [\partial]$ , the left ideal generated by $S$ is denoted by $K (x) [\partial] S$ . For instance, let $I = Q (x_{1}, x_{2}) [\partial_{1}, \partial_{2}] {\partial_{1} - 1, \partial_{2} - 1}$ . Then $I$ is D-finite because the quotient $Q (x_{1}, x_{2}) [\partial_{1}, \partial_{2}] / I$ is a vector space of dimension $1$ over $Q (x_{1}, x_{2})$ . Thus, $rank (I) = 1$ .

2.2 Gröbner bases

Gröbner bases in $K (x) [\partial]$ 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 $T (\partial)$ the commutative monoid generated by $\partial_{1}, \dots, \partial_{n}$ . An element of $T (\partial)$ is called a term. For a vector $u = (u_{1}, \dots, u_{n}) \in N^{n}$ , the symbol $\partial^{u}$ stands for the term $\partial_{1}^{u_{1}} \dots \partial_{n}^{u_{n}}$ . The order of $\partial^{u}$ is defined to be $| u | := u_{1} + \dots + u_{n}$ . For a nonzero operator $P \in K (x) [\partial]$ , the order of $P$ is defined to be the highest order of the terms that appear in $P$ effectively.

Let $≺$ ¹¹1In examples, we use the graded lexicographic ordering with $\partial_{n} ≻ \dots ≻ \partial_{1}$ . be a graded monomial ordering [10, Definition 1, page 55] on $N^{n}$ . Since there is a one-to-one correspondence between terms in $T (\partial)$ and elements in $N^{n}$ , the ordering $≺$ on $N^{n}$ induces an ordering on $T (\partial)$ with $\partial^{u} ≺ \partial^{v}$ if $u ≺ v$ . 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 $N^{n}$ in the rest of the paper.

For a nonzero element $P \in K (x) [\partial]$ , the head term of $P$ , denoted by $HT (P)$ , is the highest term appearing in $P$ . The coefficient of $HT (P)$ is called the head coefficient of $P$ and is denoted by $HC (P)$ . For a subset $S$ of nonzero elements in $K (x) [\partial]$ , $HT (S)$ and $HC (S)$ stand for the sets of head terms and head coefficients of the elements in $S$ , respectively.

For a Gröbner basis $G$ in $K (x) [\partial]$ , a term is said to be parametric if it is not divisible by any term in $HT (G)$ . The set of exponents of all parametric terms is referred to as the set of parametric exponents of $G$ and denoted by $PE (G)$ . If $K (x) [\partial] G$ is D-finite, then its rank is also called the rank of $G$ and denoted by $rank (G)$ , which is equal to $| PE (G) |$ .

2.3 Formal power series

Let $K [[x]]$ be the ring of formal power series with respect to $x_{1}, \dots, x_{n}$ . For $P \in K [x] [\partial]$ and $f \in K [[x]]$ , there is a natural action of $P$ on $f$ , which is denoted by $P (f)$ . For $P, Q \in K [x] [\partial]$ , it is straightforward to verify that

P Q (f) = P (Q (f)) .

(1)

For $u = (u_{1}, \dots, u_{n}) \in N^{n}$ , the product $(u_{1}!) \dots (u_{n}!)$ is denoted by $u!$ , and $x_{1}^{u_{1}} \dots x_{n}^{u_{n}}$ by $x^{u}$ . A formal power series can always be written in the form

f = \sum u \in N^{n} \frac{c_{u}}{u!} x^{u},

where $c_{u} \in K$ . Such a form is convenient for differentiation.

Taking the constant term $c_{0}$ of a formal power series $f$ gives rise to a ring homomorphism, which is denoted by $ϕ$ . A direct calculation yields

ϕ (\partial^{u} (f)) = c_{u} .

(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 $f \in K [[x]]$ . Then $f = 0$ if and only if, for all $u \in N^{n}$ ,

ϕ (\partial^{u} (f)) = 0.

The following result appears in [11] for $s = 1$ . But the proof applies literally also for arbitrary values of $s$ . Please see [25, Lemma 4.3.3] for a detailed verification.

Lemma 2.2.

Let $p_{1}, p_{2}, \dots, p_{s}$ and $q$ be polynomials in $K [x]$ with

gcd (p_{1}, p_{2}, \dots, p_{s}, q) = 1.

If $p_{i} / q$ has a power series expansion for each $i \in {1, 2, \dots, s}$ , then the constant term of $q$ is nonzero.

The fixed ordering $≺$ on $N^{n}$ also induces an ordering on the monoid $T (x)$ generated by $x_{1}, \dots, x_{n}$ in the following manner: $x^{u} ≺ x^{v}$ if $u ≺ v$ . A nonzero element $f \in K [[x]]$ can be written as

f = \frac{c_{u}}{u!} x^{u} + % higher monomials with respect to ≺,

where $c_{u} \in K$ is nonzero. We call $u$ the initial exponent of $f$ .

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 $I \subset K (x) [\partial]$ with rank $d$ , there exists a differential field $E$ containing $K [[x]]$ such that the solution space of $I$ has dimension $d$ over $C_{E}$ , where $C_{E}$ stands for the subfield of the constants in $E$ .

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 $E$ is a differential field described in the above proposition. For a D-finite ideal $I$ , the solution space of $I$ in $E$ is denoted by ${sol}_{E} (I)$ . Likewise, for a finite-rank Gröbner basis $G$ , the solution space of $K (x) [\partial] G$ is simply denoted by ${sol}_{E} (G)$ .

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 $V \subset E$ be a $d$ -dimensional linear subspace over $C_{E}$ . Then there exists a unique left ideal $I \subset E [\partial]$ of rank $d$ such that

V = {sol}_{E} (I) .

Furthermore, an operator $P$ belongs to $I$ if and only if $P$ annihilates every element of $V$ .

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 $K [[x]]$ are linearly independent over $K$ if and only if they are linearly independent over any field of constants that contains $K$ .

Wronskian-like determinants are expressed by elements of $T (\partial)$ via wedge notation in [19]. For $v_{1}, v_{2}, \dots, v_{ℓ} \in N^{n}$ and $ℓ \in Z^{+}$ , the exterior product

\partial^{v_{1}} \land \partial^{v_{2}} \land \dots \land \partial^{v_{ℓ}}

is defined as a multi-linear function from $E^{ℓ}$ to $E$ that maps $(z_{1}, \dots, z_{ℓ}) \in E^{ℓ}$ to:

∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \partial^{v_{1}} (z_{1}) & \partial^{v_{1}} (z_{2}) & \dots & \partial^{v_{1}} (z_{ℓ}) \partial^{v_{2}} (z_{1}) & \partial^{v_{2}} (z_{2}) & \dots & \partial^{v_{2}} (z_{ℓ}) ⋮ & ⋮ & ⋱ & ⋮ \partial^{v_{ℓ}} (z_{1}) & \partial^{v_{ℓ}} (z_{2}) & \dots & \partial^{v_{ℓ}} (z_{ℓ}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ .

It follows from Theorem 1 in [16, Chapter II] that $z_{1}$ , …, $z_{ℓ}$ are linearly independent over $C_{E}$ if there exist $v_{1}, \dots, v_{ℓ} \in N^{n}$ such that

(\partial^{v_{1}} \land \dots \land \partial^{v_{ℓ}}) (z_{1}, \dots, z_{ℓ}) \neq 0.

Let $G$ be a reduced and finite-rank Gröbner basis in $K (x) [\partial],$ the Wronskian operator of $G$ is defined to be

w_{G} := ⋀ u \in PE (G) \partial^{u} .

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

Proposition 2.5.

Let $d = rank (G)$ , $z_{1}, \dots, z_{d} \in {sol}_{E} (G)$ and $PE (G) = {u_{1}, \dots, u_{d}}$ .

The elements $z_{1}, \dots, z_{d}$ are linearly independent over $C_{E}$ if and only if $w_{G} (z_{1}, \dots, z_{d})$ is nonzero.
Let $\partial^{v}$ be the head term of an element $g$ of $G$ , and let $z_{1}, \dots, z_{d}$ be linearly independent over $C_{E}$ . Set $z = (z_{1}, \dots, z_{d})$ and

in which the elements of $T (\partial)$ are placed on the right-hand side of a product. Then

$w_{G} (z)^{- 1} (w_{G} \land \partial^{v} (z, \cdot)) = HC (g)^{- 1} g .$

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

3 Ordinary points and singularities

Let $P \in K [x] [\partial]$ be in the form $P = c_{u_{m}} \partial^{u_{m}} + c_{u_{m - 1}} \partial^{u_{m - 1}} + \dots + c_{u_{0}} \partial^{u_{0}},$ where $c_{u_{0}}, \dots, c_{u_{m}} \in K [x] ∖ {0}$ and $\partial^{u_{0}}$ …, $\partial^{u_{m}}$ are distinct. We say that $P$ is primitive if $gcd (c_{u_{0}}, c_{u_{1}}, \dots, c_{u_{m}}) = 1.$

For brevity, a Gröbner basis $G$ in $K (x) [\partial]$ is said to be primitive if it is finite, reduced and its elements are primitive ones in $K [x] [\partial]$ . Every nontrivial left ideal in $K (x) [\partial]$ 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 $G$ is a primitive Gröbner basis of finite rank. A point $α \in {¯ ¯¯ ¯ K}^{n}$ is called an ordinary point of $G$ if, for every $p \in HC (G)$ , $p (α) \neq 0$ . Otherwise, it is called a singularity of $G$ .

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

Example 3.2.

Consider the Gröbner basis in $Q (x_{1}, x_{2}) [\partial_{1}, \partial_{2}]$

G = {\partial_{2} - \partial_{1}, \partial_{1}^{2} + 1} .

We find that $HT (G) = {\partial_{1}^{2}, \partial_{2}}$ and $HC (G) = {1}$ . So $G$ has no singularity.

Example 3.3.

Consider the Gröbner basis [19, Example 3] in $Q (x_{1}, x_{2}) [\partial_{1}, \partial_{2}]$

G = {x_{1} \partial_{1}^{2} - (x_{1} x_{2} - 1) \partial_{1} - x_{2}, x_{2} \partial_{2} - x_{1} \partial_{1}} .

In this case, $HT (G) = {\partial_{1}^{2}, \partial_{2}}$ and $HC (G) = {x_{1}, x_{2}}$ and $PT (G) = {1, \partial_{1}}$ . The singularities of $G$ are

{(a, b) \in {¯ ¯¯ ¯ Q}^{2} ∣ a = 0 or b = 0},

which are two lines in ${¯ ¯¯ ¯ Q}^{2}$ . 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 $K^{n}$ can always be translated to the origin, and we may assume that $K$ is algebraically closed when necessary.

Theorem 3.4.

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

Proof.

Let $G = {G_{1}, \dots, G_{k}}$ , $\partial^{v_{i}} = HT (G_{i})$ and $ℓ_{i} = HC (G_{i})$ , $i = 1, \dots, k$ . Then we can write

G_{i} = ℓ_{i} \partial^{v_{i}} + a linear % combination of parametric terms over K [x]

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

We associate to each tuple $u \in PE (G)$ an arbitrary constant $c_{u} \in K$ . For a non-parametric term $\partial^{v}$ , let $N_{v}$ be its normal form with respect to $G$ . Although $N_{v}$ belongs to $K (x) [\partial]$ , there exists a power product $ℓ_{v}$ of $ℓ_{1}, \dots, ℓ_{k}$ such that $ℓ_{v} N_{v} \in K [x] [\partial]$ . Write

ℓ_{v} (x) N_{v} = \sum u \in PE (G) a_{u, v} (x) \partial^{u}

with $a_{u, v} \in K [x]$ . Set

c_{v} = ℓ_{v} (0)^{- 1} \sum u \in PE (G) a_{u, v} (0) c_{u} .

(3)

Note that $ℓ_{v}$ can be chosen to be any power product of $ℓ_{1}, \dots, ℓ_{k}$ such that $ℓ_{v} N_{v}$ belongs to $K [x] [\partial]$ . Let

f = \sum u \in N^{n} \frac{c_{u}}{u!} x^{u} .

We claim that $f$ is a formal power series solution of $G$ , that is,

G_{i} (f) = 0, i = 1, \dots, k .

(4)

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

ϕ (\partial^{u} G_{i} (f)) = 0

(5)

for all $u \in N^{n}$ and $i \in {1, \dots, k}$ . We proceed by Noetherian induction on the term order $≺$ .

Starting with $\partial^{0}$ , we can write

\partial^{0} G_{i} = G_{i} = ℓ_{i} (x) \partial^{v_{i}} - \sum u \in P E (G) a_{u, v_{i}} (x) \partial^{u},

(6)

where $a_{u, v_{i}} \in K [x]$ . It follows that

ℓ_{i} (x) N_{v_{i}} = \sum u \in P E (G) a_{u, v_{i}} (x) \partial^{u} .

By (3),

ℓ_{i} (0) c_{v_{i}} - \sum u \in P E (G) a_{u, v_{i}} (0) c_{u} = 0,

which can be rewritten as

ϕ (ℓ_{i} (x)) ϕ (\partial^{v_{i}} (f)) - \sum u \in PE (G) ϕ (a_{u, v_{i}} (x)) ϕ (\partial^{u} (f)) = 0.

Since $ϕ$ is a ring homomorphism, we have

ϕ ⎛ ⎝ ℓ_{i} (x) \partial^{v_{i}} (f) - \sum u \in PT (G) a_{u, v_{i}} (x) \partial^{u} (f) ⎞ ⎠ = 0.

We see that $ϕ (G_{i} (f)) = 0$ by (6).

Assume that $\partial^{v}$ is a term higher than $\partial^{0}$ and, for all $w$ with $w ≺ v$ and all $i \in {1, \dots, k}$ ,

ϕ (\partial^{w} G_{i} (f)) = 0.

Reducing $\partial^{v + v_{i}}$ modulo $G$ , we have

ℓ (x) \partial^{v + v_{i}} = p_{v} (x) (\partial^{v} G_{i}) + (\sum w ≺ v k \sum s = 1 p_{w, s} (x) (\partial^{w} G_{s})) + ℓ (x) N_{v + v_{i}},

where $ℓ (x)$ and $p_{v} (x)$ are two power products of $ℓ_{1} (x), \dots, ℓ_{k} (x)$ , and $p_{w, s} (x)$ belongs to $K [x]$ for all $w ≺ v$ and $s \in {1, \dots, k}$ . Moreover, $ℓ (x) N_{v + v_{i}}$ belongs to $K [x] [\partial]$ . Applying the above equality to $f$ , we get

ℓ (x) \partial^{v + v_{i}} (f) = p_{v} (x) (\partial^{v} G_{i}) (f) + (\sum w ≺ v k \sum s = 1 p_{w, s} (x) (\partial^{w} G_{s}) (f)) + ℓ (x) N_{v + v_{i}} (f) .

Applying $ϕ$ to the above equality yields

\begin{matrix} ϕ (ℓ (x) \partial^{v + v_{i}} (f)) & = & p_{v} (0) ϕ (\partial^{v} G_{i} (f)) + \sum_{w ≺ v} \sum_{s = 1}^{k} p_{w, s} (0) ϕ (\partial^{w} G_{s} (f)) + ϕ ((ℓ (x) N_{v + v_{i}}) (f)) . \end{matrix}

By the induction hypothesis, $ϕ (\partial^{w} G_{s} (f)) = 0$ for all $w$ with $w ≺ v$ and for all $s \in {1, \dots, k}$ . Thus,

ϕ (ℓ (x) \partial^{v + v_{i}} (f)) = p_{v} (0) ϕ (\partial^{v} G_{i} (f)) + ϕ ((ℓ (x) N_{v + v_{i}}) (f)) .

Writing $ℓ (x) N_{v + v_{i}} = \sum_{u \in PE (G)} a_{u, v + v_{i}} (x) \partial^{u}$ with $a_{u, v + v_{i}} (x) \in K [x]$ , we see that the above equality implies

ℓ (0) c_{v + v_{i}} = p_{v} (0) ϕ (\partial^{v} G_{i} (f)) + \sum u \in PE (G) a_{u, v + v_{i}} (0) c_{u} .

It follows from (3) that

p_{v} (0) ϕ (\partial^{v} G_{i} (f)) = 0.

Since $p_{v} (0)$ is nonzero, $ϕ (\partial^{v} G_{i} (f))$ is equal to zero. This proves (5). Therefore, our claim (4) holds. Since there are $rank (G)$ many parametric terms, the D-finite system $G$ has $rank (G)$ many $K$ -linearly independent formal power series solutions with initial exponents in $PE (G)$ .

Sufficiency. Let $d = rank (G)$ . Assume that $f_{1}, \dots, f_{d}$ are $K$ -linearly independently formal power series solutions of $G$ whose the initial exponents are $u_{1}$ , …, $u_{d}$ , respectively. Without loss of generality, we assume further that, for all $j, m \in {1, \dots, d}$ ,

ϕ (\partial^{u_{m}} (f_{j})) = δ_{m j},

where $δ_{m j}$ stands for Kronecker’s symbol.

Let $f = (f_{1}, \dots, f_{d})$ . By the above assumption, the constant term of $w_{G} (f)$ is nonzero. So the formal power series $w_{G} (f)$ is invertible in $K [[x]]$ .

Let $F_{i} = (w_{L} \land \partial^{v_{i}}) (f, \cdot)$ . By Proposition 2.5,

(7)

Since $G_{i}$ is primitive, we can write $G_{i}$ as

ℓ_{i} \partial^{v_{i}} + d \sum j = 1 ℓ_{i j} \partial^{u_{j}},

where $ℓ_{i j} \in K [x]$ and $gcd (ℓ_{i}, ℓ_{i 1}, \dots, ℓ_{i d}) = 1$ . By (7), we have

\frac{ℓ_{i j}}{ℓ_{i}} \in K [[x]] for each j = 1, \dots, d .

It follows from Lemma 2.2 that the constant term of $ℓ_{i}$ is nonzero. Hence, the origin is an ordinary point of $G$ . ∎

The proof for the necessity of the above theorem also holds for an arbitrary left (not necessarily D-finite) ideal $K (x) [\partial] G$ , provided that the origin is an ordinary point of $G$ . 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 $G$ be a primitive Gröbner basis of rank $d$ .

Assume that the origin is a singularity of $G$ . We call the origin an apparent singularity of $G$ if $G$ has $d$ linearly independent formal power series solutions over $K$ .
Assume that $M$ is a primitive Gröbner basis of finite rank. We call $M$ a left multiple of $G$ if

$K (x) [\partial] M \subset K (x) [\partial] G .$

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

Example 4.2.

The solution space ${sol}_{E} (G)$ of the Gröbner basis

G = {x_{2} \partial_{2} + \partial_{1} - x_{2} - 1, \partial_{1}^{2} - \partial_{1}}

in $K (x_{1}, x_{2}) [\partial_{1}, \partial_{2}]$ is generated by ${exp (x_{1} + x_{2}), x_{2} exp (x_{2})}$ . In this case,

HT (G) = {\partial_{2}, \partial_{1}^{2}}, HC (G) = {x_{2}, 1} and PE (G) = {(0, 0), (1, 0)} .

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

Let $M$ be another Gröbner basis such that

K (x) [\partial] \cdot M = K (x) [\partial] G \cap K (x) [\partial] \cdot {x_{1} \partial_{1} - 1, \partial_{2}} .

We find that $M$ is a left multiple of $G$ with rank $3$ .

Example 4.3.

The solution space ${sol}_{E} (G)$ of the Gröbner basis

G = {x_{2}^{2} \partial_{2} - x_{1}^{2} \partial_{1} + x_{1} - x_{2}, \partial_{1}^{2}}

in $K (x_{1}, x_{2}) [\partial_{1}, \partial_{2}]$ is generated by ${x_{1} + x_{2}, x_{1} x_{2}}$ . In this case,

HT (G) = {\partial_{2}, \partial_{1}^{2}}, H C (G) = {x_{2}^{2}, 1} and PE (G) = {(0, 0), (1, 0)} .

Therefore, the origin is an apparent singularity of $G$ .

Set

S = {(0, 0), (0, 1), (2, 0), (0, 2)} .

Let $M$ be another Gröbner basis with

We find that $r a n k (M) = 6$ . By Definition 4.1 (ii), $M$ is a left multiple of $G$ with $rank (M) = 6$ .

For a subset $S$ of $K (x) [\partial]$ , we denote by ${IE}_{0} (S)$ the set of initial exponents of nonzero elements in ${sol}_{E} (S) \cap K [[x]]$ and call it the set of initial exponents of $S$ at the origin. Then $| {IE}_{0} (S) |$ is the dimension of ${sol}_{E} (S) \cap K [[x]]$ , because any set of formal power series with distinct initial exponents are linearly independent over $K$ . For a primitive Gröbner basis $G$ , the origin is an ordinary point of $G$ if and only if ${IE}_{0} (G) = PE (G)$ by Theorem 3.4, it is an apparent singularity if and only if ${IE}_{0} (G) \neq PE (G)$ but $| {IE}_{0} (G) | = | PE (G) |$ 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 $I$ and $J$ be $D$ -finite ideals in $K (x) [\partial]$ . Then

$rank (I \cap J) + rank (I + J) = rank (I) + rank (J) .$
${sol}_{E} (I \cap J) = {sol}_{E} (I) + {sol}_{E} (J)$ .

Proof.

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

\begin{matrix} ψ : & V / (U \cap W) & \to & V / U \times V / W v + U \cap W & \mapsto & (v + U, - v + W), \end{matrix}

and

\begin{matrix} ϕ : & V / U \times V / W & \to & V / (U + W) (a + U, b + W) & \mapsto & a + b + (U + W) . \end{matrix}

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..

0 \to V / (U \cap W) ψ \to V / U \times V / W ϕ \to V / (U + W) \to 0.

It follows that $V / U \times V / W$ is linearly isomorphic to $V / (U \cap W) \oplus V / (U + W) .$ In particular,

dim (V / (U \cap W)) + dim (V / (U + W)) = dim (V / U) + dim (V / W) .

Setting $V = K (x) [\partial]$ , $U = I$ and $W = J$ , 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 ${sol}_{E} (I) + {sol}_{E} (J) \subset {sol}_{E} (I \cap J)$ . On the other hand,

\begin{matrix} dim ({sol}_{E} (I) + {sol}_{E} (J)) & = & dim ({sol}_{E} (I)) + dim ({sol}_{E} (J)) - dim ({sol}_{E} (I) \cap {sol}_{E} (J)) . = & dim ({sol}_{E} (I)) + dim ({sol}_{E} (J)) - dim ({sol}_{E} (I + J)) (since {sol}_{E} (I) \cap {sol}_{E} (J) = {sol}_{E} (I + J)) = & dim ({sol}_{E} (I \cap J)) (by the second % assertion). \end{matrix}

Hence, ${sol}_{E} (I \cap J) = {sol}_{E} (I) + {sol}_{E} (J)$ . ∎

As a matter of notation, we define $N_{m}^{n} = {u \in N ∣ | u | = m}$ for $m \in N$ . The second lemma illustrates a connection between parametric exponents and initial ones.

Lemma 4.5.

Let $M$ be a primitive Gröbner basis. Assume that ${sol}_{E} (M)$ has a basis in $K [[x]]$ and ${IE}_{0} (M) = N_{m}^{n}$ for some $m \in N$ . Then ${IE}_{0} (M) = PE (M)$ . Consequently, the origin is an ordinary point of $M$ .

Proof.

Assume that $f_{1}, \dots f_{ℓ} \in K [[x]]$ form a basis of ${sol}_{E} (M)$ and their initial exponents are distinct. Then $ℓ = | N_{m}^{n} |$ . Let $f = (f_{1}, \dots, f_{ℓ})$ . And set

w = ⋀ u \in {IE}_{0} (M) \partial^{u} .

Then $w (f)$ is a nonzero element in $K [[x]]$ . For every $v \in N^{n}$ with $| v | = m + 1$ , let $F_{v} = (w_{M} \land \partial^{v}) (f, \cdot)$ , which belongs to $K [[x]] [\partial]$ . Then $HT (F_{v}) = \partial^{v}$ because $w (f)$ is nonzero and the ordering $≺$ is graded. Since $F_{v}$ annihilates $f_{1}, \dots, f_{ℓ}$ , it vanishes on ${sol}_{E} (M)$ . It follows from Proposition 2.4 that $F_{v}$ belongs to the extended ideal $E [\partial] \cdot M$ , in which $M$ is still a Gröbner basis. Thus, $F_{v}$ can be reduced to zero by $M$ . Accordingly, $\partial^{v}$ is not a parametric derivative of $M$ . In other words, $PE (M)$ is a subset of $N_{m}^{n}$ . Hence, $PE (M) = N_{m}^{n}$ because $| PE (M) | = ℓ$ and $ℓ = | N_{m}^{n} |$ . The origin is an ordinary point by Theorem 3.4. ∎

Theorem 4.6.

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

Proof.

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

Set $m = {max}_{u \in {IE}_{0} (G)} | u | .$ For every $v = (v_{1}, \dots, v_{n}) \in N^{n}$ , we denote by $I_{v}$ the left ideal generated by $x_{1} \partial_{1} - v_{1},$ …, $x_{n} \partial_{n} - v_{n}$ in $K (x) [\partial]$ . Note that the solution space of $I_{v}$ is spanned by $x^{v}$ . Set

I = K (x) [\partial] G and J = ⋂ v \in N_{m}^{n} ∖ {IE}_{0} (G) I_{v} .

Then the two left ideals $I$ and $J$ have no solution in common except the trivial one because $v \in N_{m}^{n} ∖ {IE}_{0} (G)$ . It follows from Lemma 4.4 (iii) that

{sol}_{E} (I \cap J) = {sol}_{E} (I) \oplus {sol}_{E} (J) .

In particular, ${sol}_{E} (I \cap J)$ has dimension $| N_{m}^{n} |$ over $C_{E}$ , because ${sol}_{E} (I)$ and ${sol}_{E} (J)$ have dimensions $| {IE}_{0} (G) |$ and $| N_{m}^{n} | - | {IE}_{0} (G) |$ , respectively. So ${IE}_{0} (I \cap J) = N_{m}^{n}$ . Let $M$ be a primitive Gröbner basis of $I \cap J$ . Then the origin is an ordinary point of $M$ by Lemma 4.5.

Necessity. Assume that $M$

Apparent Singularities of D-finite Systems

Authors

Laurent Series Solutions of Algebraic Ordinary Differential Equations

Formal Power Series Solutions of First Order Autonomous Algebraic Ordinary Differential Equations

Univariate Contraction and Multivariate Desingularization of Ore Ideals

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package

Formal Solutions of Completely Integrable Pfaffian Systems With Normal Crossings

Continuous Ordinary Differential Equations and Ordinal Time Turing Machines

1 Introduction

2 Preliminaries

2.1 Rings of differential operators

2.2 Gröbner bases

2.3 Formal power series

Lemma 2.1.

Lemma 2.2.

2.4 Solutions and Wronskians

Proposition 2.3.

Proposition 2.4.

Proposition 2.5.

3 Ordinary points and singularities

3.1 Ordinary points

Definition 3.1.

Example 3.2.

Example 3.3.

3.2 Characterization of ordinary points

Theorem 3.4.

Proof.

4 Apparent singularities

Definition 4.1.

Example 4.2.

Example 4.3.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Theorem 4.6.

Proof.

Apparent Singularities of D-finite Systems

Authors

Related Research

Laurent Series Solutions of Algebraic Ordinary Differential Equations

Formal Power Series Solutions of First Order Autonomous Algebraic Ordinary Differential Equations

Univariate Contraction and Multivariate Desingularization of Ore Ideals

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package

Formal Solutions of Completely Integrable Pfaffian Systems With Normal Crossings

Continuous Ordinary Differential Equations and Ordinal Time Turing Machines

This week in AI

1 Introduction

2 Preliminaries

2.1 Rings of differential operators

2.2 Gröbner bases

2.3 Formal power series

Lemma 2.1.

Lemma 2.2.

2.4 Solutions and Wronskians

Proposition 2.3.

Proposition 2.4.

Proposition 2.5.

3 Ordinary points and singularities

3.1 Ordinary points

Definition 3.1.

Example 3.2.

Example 3.3.

3.2 Characterization of ordinary points

Theorem 3.4.

Proof.

4 Apparent singularities

Definition 4.1.

Example 4.2.

Example 4.3.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Theorem 4.6.

Proof.