Formal Power Series Solutions of Algebraic Ordinary Differential Equations

In this paper, we consider nonlinear algebraic ordinary differential equations (AODEs) and study their formal power series solutions. Our method is inherited from Lemma 2.2 in [J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Mathematische Annalen, 267(1984), 213-238] for expressing high order derivatives of a differential polynomial via their lower order ones. By a careful computation, we give an explicit formula for the expression. As an application, we give a method for determining the existence of a formal power series solution with given first coefficients. We define a class of certain differential polynomials in which our method works properly, which is called non-vanishing. A statistical investigation shows that many differential polynomials in the literature are non-vanishing.

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1 Introduction

Let be an algebraically closed field of characteristic zero. An algebraic ordinary differential equation (AODE) is an equation of the form

for some and a polynomial in variables over . This paper addresses a study of formal power series solutions, i.e. solutions in , of nonlinear AODEs.

The problem of finding formal power series solutions of AODEs has a long history and it has been heavily studied in the literature. The Newton polygon method is a well-known method developed for studying this problem. In [BB56], Briot and Bouquet use the Newton polygon method for studying the singularities of first-order and first degree ODEs. Fine gave a generalization of the method for arbitrary order AODEs in [Fin89]. By using Newton polygon method, one can obtain interesting results on a larger class of series solutions which is called generalized formal power series solutions, i.e. power series with real exponents. In [GS91], Grigoriev and Singer proposed a parametric version of the Newton polygon method and use it to study generalized formal power series solutions of AODEs. A worth interpretation of the parametric Newton polygon method can be found in [CF09] and [Can05]. However, it has been shown in [DRJ97] that Newton polygon method for AODEs has its own limits in the sense that in some cases, it fails to give a solution.

In this paper, we follow a different method which is inherited from the work by Denef and Lipshitz in [DL84]. In [DL84] the authors presented a possibly algorithm for determining the existence of a formal power series solution of a system of AODEs. One of the fundamental steps in their construction is the expression of  in terms of lower order AODEs for arbitrary natural numbers (see [DL84, Lemma 2.2]). In this paper, by a careful computation, we present an explicit formula for the expression (Theorem 3.6).

As a nice application, we use the explicit formula to decide the existence of formal power series solutions of a given AODE whose first coefficients are given, and in the affirmative case, compute all of them (Theorem 4.5 and 5.6). If the AODE is given together with a suitable initial value data, or equivalently suitable first coefficients, then one can decide immediately the existence of a formal power series solution and determine such a solution by Cauchy method (Proposition 2.3). However, there are formal power series solutions which cannot be determined by Cauchy method. The method we developed in this paper can be used to overcome this difficulty. Moreover, we can determine all (truncated) formal power series solutions of a certain class of AODEs.

Another interesting application of the method is to give an explicit statement of a result by Hurwitz in [Hur89]. Hurwitz proved that if is a formal power series solution of an AODE, then there exists a large enough positive integer such that coefficients of order are determined by a recursion formula. Our result (Algorithm 4.6 and 5.7) can be used to determine a sharp upper bound for such an and the recursion formula (compare with [Hoe14]).

The rest of the paper is organized as follow. Section 2 is to give necessary notations and definitions for the rest of the paper. We give a refinement for a lemma by Denef and Lipshitz in [DL84] in Section 3. Section 4 and 5 present an application of Theorem 3.6 to the problem of deciding the existence of a formal power series with given first coefficients. In Section 5, we define certain AODEs in which our method can work properly. They are called non-vanishing AODEs. Section 6 is devoted for statistical investigation of non-vanishing AODEs in the literature.

2 Preliminaries

Let be an algebraically closed field of characteristic zero and . Assume that  () is the ring of differential polynomials in with coefficients in the ring of polynomials (the field of rational functions , respectively), where the derivation of is . A differential polynomial is of order if the -th derivative  is the highest derivative appearing in it.

Consider the algebraic ordinary differential equation (AODE) of the form

(1)

where is a differential polynomial in of order . For simplicity, we may also write (1) as , and call the order of (1). As a matter of notation, we set

(2)

Next, we recall a lemma concerning the -th derivative of with respect to (see [Rit50, page 30]).

Lemma 2.1.

Let be a differential polynomial of order . Then for each , there exists a differential polynomial of order at most such that

(3)

where is the separant of .

Let be the ring of formal power series with respect to . For each formal power series  and , we use the notation to refer the coefficient of in . The coefficient of in a formal power series can be translated into the constant coefficient of its -th formal derivative, as stated in the following lemma (see [KP10, Theorem 2.3, page 20]).

Lemma 2.2.

Let and . Then .

Let be a differential polynomial of order  and be a tuple of indeterminates or elements in . As a notation we set .

Assume that is a formal power series solution of the AODE  at the origin, where is unknown. Set . By Lemma 2.2, we know that if and only if for each .

Based on the above fact and Lemma 2.1, we have the following proposition.

Proposition 2.3.
111This proposition is sometimes called Implicit Function Theorem for AODEs as a folklore.

Let be a differential polynomial of order . Assume that satisfies and . For , set

where is specified in Lemma 2.1. Then is a solution of .

In the above proposition, if the initial value vanishes at the separant of , we may expand in Lemma 2.1 further to get formal power series solutions, as the following example illustrates.

Example 2.4.

Consider the following AODE:

Since , we cannot apply Proposition 2.3 to get a formal power series solutions of the above AODE. Instead, we observe from computation that

(4)

where is of order and .

Assume that is a formal power series solution of the AODE , where is to be determined. From , we have that .

If we take , then we can deduce from (4) that for each ,

Thus,

where . Therefore, we derive a formal power series solution of with .

If we take , then we observe that

It implies that there is no constraint for in the equation . For , it follows from (4) that

Thus,

where . Therefore, we derive formal power series solutions of with and is an arbitrary constant in .

In the above example, we expand to the second highest derivative so that there are some non-vanishing coefficients to compute all formal power series solutions of recursively. In the next sections, we will develop this idea in a systematical way.

3 Refinement of Denef-Lipshitz Lemma

In [DL84][Lemma 2.2], given a differential polynomial , the authors present an expansion formula for with arbitrary . In this section, by a careful analysis, we refine the expansion and give an explicit formula (see Theorem 3.6). This new formula plays a fundamental role in the computation of formal power series in next sections.

Lemma 3.1.

Let be a differential polynomial of order . Then we have

(5)

where .

Proof.

Assume that . Then

Thus,

where

and

If , then it follow from (2) that both sides of (5) are equal to . ∎

Proposition 3.2.
222We thank Christoph Koutschan for bring this proposition to our attention, which leads to a short proof of Corollary 3.3.

Let be a differential polynomial of order . Then for each , we have

(6)

where .

Proof.

We use induction on to prove the above claim.

Let . Then for each , the left side of (6) is . And the right side is

Assume that the claim holds for . We consider the case. If , then the left side of (6) is . And the right side is . It follows from Lemma 2.1 that the claim holds for .

If , the it follows from Lemma 3.1 that

From induction hypothesis, we have that

Similarly,

Therefore, we have that

Corollary 3.3.

Let be a differential polynomial of order . Then for each , we have

Proof.

In Proposition 3.2, we set , and . ∎

Proposition 3.4.

Let be a differential polynomial of order . Then for each , we have

where

  1. for ;

  2. is a differential polynomial of order at most .

Proof.

We use induction on to prove the above claim.

Let . By Lemma 2.1, we have

where and is a differential polynomial of order . Set and .

Assume that the claim holds for . Then

Therefore,

By Lemma 2.1, we have

(7)
(8)

where are of order at most , .

Set

From the last formula of , (7), (8) and the above two formulas, we have

Next, we show that . By the definition of , we see that

By the induction hypothesis, we have that

Using the above formula, it follows that

By the induction hypothesis, we have

By Corollary 3.3, we conclude that

The following theorem is the main result of this section. This theorem can be considered as a refinement of [DL84, Lemma 2.2].

Theorem 3.5.

Let be a differential polynomial of order . Then for each , we have

where

  1. for ;

  2. is a differential polynomial of order at most .

Proof.

We use induction on to prove the above claim.

Let . It follows from Proposition 3.4.

Assume that the claim holds for . Then we have

Set , which is a differential polynomial of order at most . Then

Assume that is a differential polynomial of order . For each , we define

and

and

Then we can write Theorem 3.5 into the following matrix form:

Theorem 3.6.

Let be a differential polynomial of order . Then for each , we have

(9)

where is a differential polynomial of order at most .

Proof.

It follows from Theorem 3.5. ∎

Definition 3.7.

Assume that is a differential polynomial of order . For , we call the -th separant matrix of .

Note that the -th separant matrix is exactly the usual separant of .

4 Non-vanishing properties at an initial tuple

In this section, we consider the problem of deciding when a solution modulo a certain power of of a given AODE can be extended to a full formal power series solution. As a nice application of Theorem 3.6, we present a partial answer for this problem. In particular, given a certain number of coefficients satisfying some additional assumptions, we propose an algorithm to check whether there is a formal power series solution whose first coefficients are the given ones, and in the affirmative case, compute all of them (see Theorem 4.5 and Algorithm 4.6).

Let and . As a matter of notation, we set