Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_0,y_0) \in \mathbb {C}^2$$\end{document}, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(x_0)=y_0$$\end{document}.


Introduction
In [2], we have studied local solutions of first order autonomous algebraic ordinary differential equations.In this paper, we generalize the results obtained there to systems of higher order autonomous ordinary differential equations in one unknown function which associated algebraic set is of dimension one, i.e. the algebraic set is a finite union of curves and, maybe, points.In particular, we prove that every fractional power series solution of such systems is convergent, and an algorithm for computing these solutions is provided.Note that in [4] it is shown that for general systems of algebraic ordinary differential equations the existence of non-constant formal power series solutions can not be decided algorithmically.Nevertheless, in the case of systems as above, this undecidability property does not hold.
Finding rational general solutions of such systems has been studied in [8].There, a necessary condition on the degree of the associated algebraic curve is provided.If the condition is fulfilled, the solutions are constructed from a rational parametrization of a birational planar projection of the associated space curve.Here, we provide an algorithm which decides the existence of not only rational but also algebraic solutions of such systems.Differently to the method described in [8], in the current approach we do not need to consider a rational parametrization of the associated curve.We instead triangularize the given system and we derive from there a single autonomous ordinary differential equation of first order with the same non-constant formal Puiseux series solutions.We call it the reduced differential equation of the system.Since rational or algebraic functions are determined by their Puiseux series expansion, the reduced differential equation has also the same algebraic solutions that the original system.Furthermore, taking into account that the reduced equation is autonomous and of order one, we bound, using the results from [1], the degree of its possible algebraic solutions, and hence of those of the original system.Once the degree of the solutions is bounded, one may use the algorithm from section 4.3 in [1] to decide the existence and compute such solutions.
We derive the existence and convergence of formal Puiseux series solutions of such systems (see Theorem 3.5 and 3.4) from the corresponding results (see [2]) applied to the obtained reduced differential equation.With respect to the convergence of formal solutions a related result is given by Gerasimova and Razmyslov in [6].They show the convergence of formal power series solutions of a system of ordinary differential equations under some additional conditions such as that there are no zero-divisors of the differential algebra induced by the system and that the system is of transcendence degree one.Their method is based on the fact that the induced differential algebra is finitely generated as an algebra over its base field and then they reduce the problem to the Cauchy-Kowalevski theorem.Their method does not deal with fractional power series solutions.
In the literature there are several methods to triangularize differential systems and to obtain resolvent representations of them, see for instance [3,9] and references therein.The description of these methods are quite involved because they apply to general differential systems.This paper addresses only ordinary differential systems which associated algebraic set has dimension one and we can split the process into an algebraic triangularization part and then a straightforward differential elimination process.For the algebraic part we use regular chains as described in [7] and [11].This simple description of the process allow us to have a precise relation between the formal Puiseux series solutions of the original system and those of the reduced differential equation.
The structure of the paper is as follows.In Section 2 we recall some necessary concepts such as regular chains and regular zeros.Section 3 is devoted to derive from a system of autonomous ordinary differential equations of dimension one in one unknown function, a finite union of such regular chains.From them we derive a single autonomous ordinary differential equation of order one with the same non-constant formal Puiseux series solutions as the original system.Using this reduction, the main results in [2] can be generalized to these particular systems (see Theorems 3.4 and 3.5).In Section 4 we present an algorithm for this reduction and, using the algorithms in [1] and [2], all formal Puiseux series and algebraic solutions of the original system can be found as we illustrate by examples.

Preliminaries
We recall the notion of regular chains and regular zeros; for further details we refer to [7] and [11].Let us denote for f, g ∈ C[y 0 , . . ., y m ] by lv(f ) the leading variable, by lc(f ) the leading coefficient and by init(f ) the initial of f with respect to the ordering y 0 < • • • < y m .In addition, we denote by Res y i (f, g) the resultant of f and g with respect to y i .Let S = {F 1 , . . ., F M } ⊂ C[y 0 , . . ., y m ] be a finite system of polynomials in triangular form, i.e. lv(F i ) < lv(F j ) for any 1 ≤ i < j ≤ M. Then we define Res(f, S) as the resultant of f and consecutively F M , . . ., F 1 with respect to their leading variables, i.e.
Moreover, we define init(S) = {init(F j ) | 1 ≤ j ≤ M} and pinit(S) = M j=1 init(F j ).A regular chain is a system of algebraic equations S in triangular form with the additional property that Res(f, S) = 0 for any f ∈ init(S).
Let K ⊇ C be a field and S ⊂ C[y 0 , . . ., y m ].Then let us denote For a regular chain S, we define a regular zero of S as an element a = (a 0 , . . ., a m ) We recall a well-known theorem for the relation between regular chains and regular zeros, see [11] (1) S is a regular chain.
(2) |S| = 1 or for any k = 2, . . ., M the subsystem S k−1 is a regular chain and for any regular zero a of S k−1 and f ∈ S k it holds that init(f )(a) = 0.
In fact, statement (2) above is used in [7] as definition of regular chains.Note that in [12] regular chains are called "proper ascending chains", but are defined exactly as in this paper.
Regular chains can be helpful in order to represent algebraic sets as Theorem 5.2.2 in [11] shows: Theorem 2.2.Let S ⊆ C[y 0 , . . ., y m ] be a polynomial system.Then there exists a finite set of regular chains S 1 , . . ., S N ⊆ C[y 0 , . . ., y m ] such that We note that, in the notation of [11], V K (S j / init(S j )) = V K (S j ) \ V K (pinit(S j )), as it is also mentioned in chapter 1.5 therein.Let us recall that V C (S j ) is an algebraic set of dimension m − |S j |.
There are several implementations for performing computations with regular chains and in particular computing regular chain decomposition as in (2.1) such as in the Maple-package RegularChains.
Let K be the field of formal Puiseux series expanded around any x 0 ∈ C ∞ , where C ∞ = C ∪ {∞}.We are interested in non-constant formal Puiseux series solutions y(x) ∈ K such that y(x 0 ) = y 0 ∈ C ∞ .Since the systems we are dealing with, see (3.1) below, are invariant under the translation of the independent variable, we can assume without loss of generality that the formal Puiseux series is expanded around zero or at infinity such as in [2].For any subset S ⊆ C[y, y ′ , . . ., y (m) ] let us denote V K ( S) = {(a 0 , a 1 , . . ., a m ) ∈ K m+1 | F (a 0 , a 1 , . . ., a m ) = 0 for all F ∈ S}.

Systems of algebro-geometric dimension one
Let us consider systems of differential equations of the form where F 1 , . . ., F M ∈ C[y, y ′ , . . ., y (m) ] with m > 0. For a field K ⊇ C, by considering y and its derivatives as independent variables, we write V K ( S) for the algebraic set generated by S. We assume the dimension of V C ( S) to be one, i.e.V C ( S) is a finite union of curves and, maybe, a finite union of points.Note that a single AODE of order one can be seen as a system of the type (3.1) with M = m = 1 and is of dimension one.
Lemma 3.1.For every S as in (3.1) we can compute a finite union of regular chains S as in (3.2) with the same non-constant formal Puiseux series solutions.
Proof.Let us choose the ordering y < y ′ < • • • < y (m) .By Theorem 2.2 there is a regular chain decomposition S 1 , . . ., S N such that every regular chain has a zeroset of dimension zero or one.We omit systems of regular chains starting with an algebraic equation in y, since they only lead to constant solutions.Thus, the remaining systems are of dimension one and of the form (3.2) j=0 G m,j (y, . . ., y (m−1) ) • (y (m) ) j = 0 with r j ≥ 1 and init(G j ) = G j,r j = 0 for every 1 ≤ j ≤ m.
Now we want to study in (2.1) which kind of solutions might be a solution of a regular chain but not of the original system, i.e. the solutions of S and pinit(S) = init(G 1 ) • • • init(G m ) = 0.If y(x) is a non-constant formal Puiseux series solution of a S j , then (y(x), y ′ (x), . . ., y (m) (x)) is a regular zero of S j , because y(x) is transcendental over C and for every 1 ≤ k ≤ m we have Then, by Theorem 2.1, init(G 2 )(y(x), y ′ (x)), . . ., init(G m )(y(x), . . ., y (m−1) (x)) = 0.
Since init(G 1 )(y) = 0 is an algebraic equation in y, there can only be constant common zeros of S j and pinit(S j ).System (3.2) could be further decomposed into systems with the factors of G 1 as initial equations.However, for our purposes it is sufficient that G 1 and its separant, namely ∂ G 1 ∂u 1 , have no common differential solutions, i.e. if G 1 (y(x), y ′ (x)) = 0 for y(x) ∈ K then ∂ G 1 ∂u 1 (y(x), y ′ (x)) = 0. To ensure this we consider G 1 ∈ C[u 0 , u 1 ] to be square-free and with no factor in C[u 0 ] or C[u 1 ]; compare with the hypotheses in [2].
Moreover, we can assume without loss of generality for every solution y(x) ∈ K of a system S as in (3.2) that y(0 ) (j) ), and let S * be the system {G * j = 0} 1≤j≤m .In this situation, if y(x) ∈ K is a solution of S such that y(x 0 ) = ∞, then ỹ (x) = 1/y(x) is a formal Puiseux series solution of S * with ỹ (x 0 ) ∈ C.Moreover, for j > 0, the j-th derivative of ỹ can be written as ỹ (j) = −y j−1 y (j) + P j (y, . . ., y (j−1) ) y j+1 where P j ∈ C[u 0 , . . ., u j−1 ].In this situation, we consider the rational map where w 0 = 1/u 0 and Since the equality above is linear in u j , Φ is birational.In addition, taking into account that u 0 is not a factor of G 1 (u 0 , u 1 ), one has that the Zariski closure of Φ(V C (S)) is V C (S * ).Since dim(V C (S)) = 1, also dim(V C (S * )) = 1 and one may proceed with S * instead of S.
Now we recursively substitute in (3.4) the variables u 2 , . . ., u m by A 2 (u 0 , u 1 ), . . ., A m (u 0 , . . ., u m−1 ) to obtain the new rational functions B 2 , . . ., B m ∈ C(u 0 , u 1 ): (3.5) Observe that the denominators of the rational functions A j are powers of the separant and depend only on u 0 and u 1 .Finally we set H m (u 0 , u 1 ) = num(G m (u 0 , u 1 , B 2 (u 0 , u 1 ), . . ., B m (u 0 , u 1 ))), where num(f ) denotes the numerator of the rational function f .In this situation, we introduce a new autonomous first order algebraic differential equation, namely (3.7) H(y, y ′ ) = gcd(H 1 , . . ., H m )(y, y ′ ) = 0, and call it the reduced differential equation (of S).Note that by construction, H divides G 1 .Moreover, if S contains only one single equation G 1 , then the reduced differential equation of S is equal to G 1 .
Theorem 3.2.Let S be as in (3.2).Then S and its reduced differential equation have the same non-constant formal Puiseux series solutions.