Liouvillian solutions of first-order algebraic ODEs - Nguyễn Trí Đạt

MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY NGUYEN TRI DAT LIOUVILLIAN SOLUTIONS OF FIRST-ORDER ALGEBRAIC ORDINARY DIFFERENTIAL EQUATIONS DOCTORAL DISSERTATION IN MATHEMATICS BINH DINH – 2024 MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY NGUYEN TRI DAT LIOUVILLIAN SOLUTIONS OF FIRST-ORDER ALGEBRAIC ORDINARY DIFFERENTIAL EQUATIONS Speciality: Algebra and number theory Speciality code: 9 46 01 04 Reviewer 1: Prof. Phung Ho Hai Reviewer 2: Prof. Dang Duc Trong Reviewer 3: Assoc. Le Anh Vu Supervisors: 1.

Ngo Lam Xuan Chau 2. Le Cong Trinh BINH DINH – 2024 Declaration This dissertation was done at the Department of Mathematics and Statistics, Quy Nhon University under the supervision of Dr. Ngo Lam Xuan Chau and Assoc. Le Cong Trinh.

I hereby declare that the results presented in it are truthful and original. Most of them were published in peer-reviewed journals, others have not been published elsewhere. For using results from joint papers I have gotten permissions from my co-authors. Binh Dinh, 2024 Nguyen Tri Dat i Abstract Differential equations have been studied for a long time.

Various exact solution methods have been proposed for special cases. The main aim of this dissertation is to develop and investigate new methods for determining liouvillian solutions of first-order algebraic ordinary differential equations (AODEs). For this purpose, the differential problem is transformed into an algebraic geometric one by considering the differential equation to be an algebraic equation. Such an equation defines an algebraic curve and therefore, tools from algebraic geometry can be applied.

In particular, parametrizations of algebraic curves and algebraic function fields are intrinsically used to solve the problem and prove properties of the obtained solutions. A first idea for determining rational liouvillian solutions of first-order autonomous AODEs is presented. This approach is a generalization of a well-known algorithm for finding rational solutions. It admits an extension to the computation of the liouvillian solutions which is obtained by considering the wider differential fields.

A second focus lies on the extension of the first idea to the problem of finding liou- villian solutions of first-order autonomous AODEs of genus zero. In this situation, the theory of associated fields of algebraic functions is applied to prove a liouvillian solution (if there exists) must be a rational liouvillian solution. This leads to a classification of the liouvillian solutions respect to algebraic and transcendental cases. Last focus studies liouvillian solutions of first-order AODEs.

If an AODE is of genus zero, we prove that its liouvillian solutions can be found via first-order quasi- linear ODEs by means of associated fields of algebraic functions and optimal rational parametrizations. This method inherits the approach of existing algorithms for find- ing rational general solutions. Finally, we present an approach for solving first-order AODEs of positive genera by means of power transformations. ii Acknowledgments First of all I want to thank my supervisor Dr.

Ngo Lam Xuan Chau for the possibility to work at QNU and in particular in his research group. He helped me a lot to become a more independent researcher and always encouraged me to work on my own ideas. Working under his enthusiasm and kind guidance is an honor for me. Without him this dissertation could not have been finished.

I also want to express my gratitude to my co-supervisor Assoc. Le Cong Trinh for his hospitality during my research visits at QNU. Moreover, he taught me a lot from his mathematical expertise when we were working together on the seminars at QNU. I want to thank the colleagues and secretaries at the Department of Mathematics and Statistics and the Department of Graduate Training for their help and the friendly atmosphere throughout many occasions when I came and worked at QNU.

I also want to thank my own institute, UTH, for letting me a chance to obtain the Doctor’s degree. I want to express my gratitude to the coaches of Duong-Sinh-Tam-The club at Xuan Yen Ward, Song Cau Town for giving the cure to my severe illness and teaching me how to nurture my spiritual and physical health, that I can overcome my hard time and continue my work. My special thanks are due to my family – especially my wife Tuyet Phuong and my son Minh Tien – for their love, understanding and supporting throughout the years, that I was able to focus on my research. Binh Dinh, 2024 Nguyen Tri Dat iii Contents Introduction 1 1 Preliminaries 5 1.2 Plane algebraic curves .3 Fields of algebraic functions of one variable .4 Rational functions on algebraic curves .1 Associated fields of algebraic functions.

19 2 Rational liouvillian solutions of first-order autonomous AODEs 21 2.1 Solving first-order AODEs by parametrizations .2 Rational liouvillian solutions .4 An algorithm and examples. 33 3 Liouvillian solutions of first-order autonomous AODEs of genus zero 39 3.3 An algorithm and applications. 45 iv 4 Liouvillian solutions of first-order AODEs 54 4.1 Liouvillian solutions of first-order AODEs of genus zero .1 Associated differential equations .2 Main results and an algorithm .3 An investigation of first-order ODEs (4.2 Power transformations and their applications .2 Reduced forms by power transformations .3 Möbius transformations .4 Liouvillian solutions of first-order AODEs with liouvillian coefficients. 75 Index 79 Bibliography 82 Curriculum vitae 87 v List of algorithms Page Algorithm RatSol Rational general solutions of first-order autonomous AODEs 24 Algorithm RatLiouSol Rational liouvillian solutions of first-order autonomous AODEs 33 Algorithm LiouSolAut Liouvillian solutions of first-order autonomous AODEs of genus zero 45 Algorithm LiouSol Liouvillian solutions of first-order AODEs of genus zero 57 Algorithm RedPol Reduced forms of irreducible polynomials 67 vi Table of notations k : A differential field of characteristic zero K : The field of constants of a differential field k E : The differential extension field of k K(x) : The differential field of rational functions in x with constants K Q : The algebraic closure field of rational numbers C : The field of complex numbers K : An algebraic closed field of characteristic zero A2 (K) : The affine plane over K P2 (K) : The projective plane over K K[t]; K[x, y] : The polynomial ring of one; two variables over K K(x){y} : The ring of differential polynomials in y over K F : A polynomial in K[x, y] or differential polynomial in K(x){y} Σ : A prime differential ideal Σi : Essential prime differential ideals {F } : The radical differential ideal generated by the differential polynomial F SF : The separant of the differential polynomial F degx G : The degree of G with respect to x V(F ) : The zero set of F ∈ K[x, y] in A2 (K) C = V(F ) : The affine algebraic curve defined by F F̂ : The homogenization of F ∈ K[x, y] Γ = V(F̂ ) : The projective closure of the affine algebraic curve C g(Γ) : The genus of the projective curve Γ R(P2 ) : The field of rational functions on P2 (K) R(A2 ) : The field of rational functions on A2 (K) R(F̂ ) : The field of rational functions on F̂ E; L; K(η, ξ) : The field of algebraic functions of one variable K(t) : The field of rational functions of one variable rest (A, B) : The Sylvester resultant respect to A, B ∈ K[t]\{0} exp x : The exponent of x log x : The inverse function of exp x o : The valuation ring p : The place (of a valuation ring) gL : The genus of the function field L vii Introduction A differential equation (DE) is an equation that includes one or more unknown functions and their derivatives.

The history of DEs can be traced back to the inven- tion of calculus by Newton (in physics) and Leibniz (in pure mathematics) around 1660s–1670s. In application, the functions generally represent physical quantities, the derivatives represent their rates of change, and the DE defines a relationship between the two. Hence, these DEs play a prominent role in many disciplines including physics, engineering, economics, and biology. If a DE contains an unknown function and its derivatives which depend on an independent variable x then it is called an ordinary differential equation (ODE).

A DE is called linear if the relationship of the unknown function and its derivatives is linear; otherwise, it is called nonlinear. Such DEs can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Unfortunately, there are very few methods of solving nonlinear DEs exactly. Most ODEs encountered in physics are linear; hence, there are many ways for solving them.

An idea of transforming nonlinear DEs into linear DEs and then solve the last ones may be a reasonable candidate. However, it works for only some cases. Therefore, studying independently the solutions of nonlinear DEs is necessary, and it also contains a lot of challenges. In this dissertation, we study liouvillian general solutions of first-order algebraic ordinary differential equations (AODEs) which is a fundamental problem in the theory of non-linear algebraic DEs.

A first-order AODE is a DE of the form F (y, y ′ ) = 0, where F is an irreducible polynomial in two variables with coefficients in K(x), K is an algebraically closed field of characteristic zero. Solving an AODE is a problem of determining differentiable functions y = y(x) satisfying F (y(x), y ′ (x)) = 0. an algebraic extension field of K(x)), then it is called a rational solution (resp. an algebraic solution).

If such a solution y(x) belongs to a liouvillian extension of K(x), then it is called a liouvillian solution. A solution may contain an arbitrary constant. In this case, such a solution is called a general solution. For example, y(x) = exp(x2 + c) is a liouvillian general solution of the first-order AODE y ′ − 2xy = 0.

1 First-order AODEs have been studied a lot and there are many solution methods for their special classes. The study of these AODEs can be dated back to the works of Fuchs [16] (1884). In [20] (1926), Ince presented an overall picture of ODEs. In [30,31] (1970s), Matsuda classified differential function fields having no movable critical points up to isomorphism of differential fields.

By focusing on particular solutions, in [29] (1913), Malmquist studied the class of first-order AODEs having transcendental meromorphic solutions, and Eremenko revisited later in [10] (1982). Applied Matsuda’s theory, Eremenko in [11] (1998) gave a theoretical consideration on a degree bound for rational solutions which sheds light on the issue of finding the solution’s explicit form. Finding the closed form solution of an ODE can be traced back to the works of Liouville (1830s) for the simplest ODE y ′ = α, where α ∈ k and k is a differential field of characteristic zero. If such an equation has a solution in some elementary differential extension field E of k having the same subfield of constants K, then there exist constants c1 , c2 ,.

, un ∈ Kk and v ∈ k such that n X u′ α= ci i + v ′. i=1 ui In [44] (1968), Rosenlicht showed how Liouville theorem can be handled algebraically. For the algorithm consideration of such ODE, the pioneer work is due to Risch. In R [41, 42] (1960s), Risch described a method to determine an elementary integral u where u is an elementary function.

To extend Risch’s method, in [51, 52] (1970s), Singer studied elementary solutions of first-order AODEs. As a special result, there are necessary and sufficient conditions for the ODE y ′ = R(y) ∈ C(y) having an elementary solution. In [56] (2017), Srinivasan generalized this result to the case of liouvillian solutions with the same conditions. In [25] (1986), Kovacic presented an effective method to find liouvillian solutions of second order linear homogeneous ODEs.

This work contains an algorithm for finding rational general solutions of a Riccati equation which is applicable to the works of Chen and Ma [7] (2005) and Vo et al.