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MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY HA TRONG THI ALGEBRAIC SOLUTIONS OF SOME CLASSES OF FIRST-ORDER ALGEBRAIC ORDINARY DIFFERENTIAL EQUATIONS SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Binh Dinh - 2021 MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY HA TRONG THI ALGEBRAIC SOLUTIONS OF SOME CLASSES OF FIRST-ORDER ALGEBRAIC ORDINARY DIFFERENTIAL EQUATIONS Speciality : Algebra and Number Theory Code 9460104 : Reviewer : Prof Dr.Sc Phung Ho Hai Reviewer : Ass Prof Dr Truong Cong Quynh Reviewer : Ass Prof Dr Mai Hoang Bien Supervisors: DR NGO LAM XUAN CHAU DR LE THANH HIEU Binh Dinh - 2021 i Declaration This thesis was completed at the Department of Mathematics and Staticstics, Quy Nhon University, under the supervision of Dr Ngo Lam Xuan Chau and Dr Le Thanh Hieu I hereby declare that the results presented in this thesis are new and original Most of them were published in peer-reviewed journals, and others have not been published elsewhere For using results from joint papers I have gotten permissions from my co-authors Quy Nhon, 02 November, 2021 Author Ha Trong Thi ii Acknowledgments iii Contents Preface 1 Preliminary 1.1 Background on algebra 1.2 Differential algebra 1.3 Rational algebraic curves Equivalence transformations on first-order AODEs 2.1 Equivalence transformations 2.2 Măobius transformations Algebraic solutions of first-order algebraic ordinary differential equations 3.1 Algebraic solutions 3.2 The existence of the solutions under group transformations 3.3 A degree bound for algebraic general solutions 10 The equivalence of rationally parametrizable first-order algebraic ordinary differential equations 11 4.1 11 Polynomial ordinary differential equations 4.1.1 Differential invariant under the transformation y = z + b 11 4.1.2 Differential invariant under the transformation z = aw 4.1.3 Differential invariant under the transformation y = aw + b 12 12 4.2 Riccati differential equations 12 4.3 Abel’s differential equations 13 iv 4.4 Rationally parametrizable first-order algebraic ordinary differential equations 4.5 15 Algebraic general solutions of autonomous rationally parametrizable differential equations 17 Conclusion 19 List of author’s papers related to the thesis 20 REFERENCES 21 PREFACE A first-order algebraic ordinary differential equation (shortly, first-order AODE ) over C(x) is of the form F (y, y ) = 0, where F ∈ C(x)[y, y ] effectively contains the derivative y If F ∈ C[y, y ], then we say the equation F (y, y ) = is autonomous (i.e all coefficients of F are constants) First-order AODEs have been studied since the late nineteenth century and early twentieth century by the works of L Fuchs [14], H Poincar´e [27] and ∂ F (y, y ) = J Malmquist [19] A common solution of F (y, y ) = and ∂y is called a singular solution Singular solutions of F (y, y ) = are always algebraic solutions, and there are finitely many singular solutions Although these singular solutions are easily computed, determining whether the equation F (y, y ) = has algebraic general solutions or not and finding an algorithm for computing an algebraic general solution are difficult problems The works in [22, 23] use Măobius transformations to show that there is a class of non-autonomous first-order AODEs which are equivalent to autonomous one having algebraic general solutions The theoretical study on this problem is hence reasonable In addition, based on a degree bound of non-trivial algebraic solutions of a first-order autonomous algebraic ordinary differential equation, we can deduce a degree bound for the algebraic general solution Extending this to non-autonomous AODEs is an open question that needs to be solved A solution of the first-order algebraic ordinary differential equation F (y, y ) = in a differential field extension K of C(x) is an element η ∈ K such that F (η, η ) = 0, where “ ” is the derivation of K extending the usual derivation of C(x) If F is of degree one in y , then the differential equation F (y, y ) = is usually written in the rational form y = P (x, y)/Q(x, y), where P and Q are relatively prime polynomials in two variables The well-known Poincar´e problem is finding a degree bound for algebraic solutions of this rational differential equation In 1994, Carnicer [4] solved this problem in the non-dicritical case In 1998, Eremenko [12] showed that there is an effective degree bound for rational solutions of the differential equation F (y, y ) = and it is a challenging question whether this work can be extended to algebraic solutions More recently, Feng et al [13, 2] have provided a degree bound for algebraic general solution of autonomous AODEs of order one In addition, computing an algebraic general solution of these equations amounts to computing a nontrivial algebraic solution Computing explicit rational solutions and algebraic solutions of non-autonomous first-order AODEs has been attracting domestic and oversea mathematicians during the past few years The approach of R Feng and X-S Gao [13] in the autonomous case can be extended in a natural way to parametrizable non-autonomous class in several papers of L X C Ngo and F Winkler [22, 25, 26] These problems were more completely studied in N T Vo’s Ph.D thesis [34] with new algorithms for computing rational solutions of these equations Apart from solving first-order AODEs, classifying the equivalences of such equations is also of interest In [23, 24, 21], the authors have provided different equivalence relations on first-order AODEs According to this, solving an algebraic ordinary differential equation can amount to one in the same equivalence class Another question is to classify equations concerning such equivalence relations This thesis aims to seek some classes of first-order AODEs, in which we can determine whether an algebraic general solution exists In case such a solution exists, we provide an algorithm to compute this solution In particular, the thesis focuses on studying the existence and computing an algebraic general solution of first-order AODEs, which is equivalent to an autonomous equation and rationally parametrizable algebraic ordinary differential equation Apart from Preface, Conclusion, List of notation, References, List of the author’s publications, this thesis consists of four chapters: Chapter presents some basic notions and results which will be used in the latter chapters, including basic results in algebra, differential algebra, and rational algebraic curves Chapter gives a brief overview of equivalence transformations on AODEs of order one and studies the equivalence transformation with respect to a Măobius transformation We provide an invariant property of the differential total degree of AODEs of order one Chapter provides some properties of the solutions of AODEs which are invariant under the group actions of Măobius transformations In particular, algebraic general solutions are preserved Associated with the invariant property of differential total degree, we give a degree bound for AODEs of order one in an autonomous class Then we propose an algorithm to compute algebraic general solutions of different equations in the autonomous equivalence class Chapter gives a criterion for the equivalence of polynomial differential equations in the form y = P (x, y) As a consequence, we propose an algorithm to check whether two first-order algebraic differential equations that are rationally parametrizable are in the same equivalence class Finally, we offer another algorithm for computing algebraic general solutions of first-order AODEs that are rationally parametrizable and in an autonomous equivalence class Chapter Preliminary 1.1 Background on algebra This section lists some basic notions in the field of algebra, which are referred from [18, 8] 1.2 Differential algebra In this section, we present a background on differential rings, differential fields, and differential ideals These concepts are used to construct the notions of general solution, singular solution of differential equations The content of this section is followed from [30, 16, 3] 1.3 Rational algebraic curves This section is devoted to present some basic notions and important results in the theory of algebraic curves, which will be used in Chapter The main content of this part is referred to [32] Theorem 3.6 ([2]) Let F ∈ Q[y, y ] be an irreducible polynomial on Q Suppose that P ∈ Q[x, y] is irreducible and P (x, y) = is a non-trivial algebraic solution of the autonomous differential equation F (y, y ) = Then degy P ≤ degy F + degy F degx P = degy F, Furthermore, P (x + c, y) = is an algebraic general solution of F (y, y ) = and this degree bound is fine, i.e there exists an autonomous equation F (y, y ) = such that the equality occurs in the above degree bound 3.2 The existence of the solutions under group transformations (1) Theorem 3.7 Let F, G ∈ AODE K and suppose that F ∼ G Then F has an algebraic general solution if and only if so does G Definition 3.8 For a constant c ∈ C, we define the translation map (1) (1) Tc : AODE K → AODE K (1) defined by Tc F = F (x + c, y, y ) for all F ∈ AODE K (1) Proposition 3.9 F ∈ AODE K is autonomous if and only if Tc F = F for all c ∈ C (1) Definition 3.10 Let F ∈ AODE K be autonomous and ΦM be a transformation such that ΦM • F is autonomous An algebraic solution P (x, y) = of F (y, y ) = over C(x) is said to be non-trivial with respect to ΦM if degx (ΦM • P ) > Theorem 3.11 Let F (y, y ) = be a first-order algebraic ordinary differential equation which is autonomous, and ΦM be a transformation such that ΦM •F = is autonomous Suppose that P (x, y) = is a non-trivial algebraic solution of F (y, y ) = over C(x) with respect to ΦM Then ΦM −1 • (TC (ΦM • P )) = is an algebraic general solution of F (y, y ) = for an arbitrary constant C 10 Theorem 3.12 Suppose that F (y, y ) = is in the autonomous class and P (x, y) = is a non-trivial algebraic solution of F (y, y ) = with respect to ΦM Then the genus of the algebraic curve P (x, y) = is the same as the genus of the algebraic curve F (y, y ) = 3.3 A degree bound for algebraic general solutions (1) (1) Theorem 3.13 Let F ∈ AODE K and suppose that there exists ΦM ∈ GK such that ΦM • F is an autonomous AODE Then the total degree of an algebraic general solution of F (y, y ) = over K is bounded above by (δF +degy F ) ay + b and that the degrees of a, b, c, d are In addition, suppose that M (y) = cy + d less than N Then the minimal polynomial of the algebraic general solution of F (y, y ) = has the degree in x less than degy F + N (δF + degy F ) Algorithm Input: F ∈ K[y, y ], degy F > 0, degy F > 0, M (y) = ΦM • F is autonomous ay + b such that cy + d Output: Compute an algebraic general solution of F = if exists Applying Algorithm 4.4 in [2] to compute a non-trivial algebraic solution of ΦM • F If NOT exist, RETURN “F = has no algebraic general solution” Else, go to step 2 Let Q(z, y) = be the non-trivial algebraic solution of ΦM • F = RETURN (−cy + a)degy Q Q(z + C, M −1 (y)) = is an algebraic general solution of F = for arbitrary constant C 11 Chapter The equivalence of rationally parametrizable first-order algebraic ordinary differential equations 4.1 Polynomial ordinary differential equations A polynomial ordinary differential equation is in the form y = an (x)y n + an−1 (x)y n−1 + · · · + a1 (x)y + a0 (x), (4.1) where a0 , a1 , , an ∈ K, an = 4.1.1 Differential invariant under the transformation y = z + b Equation (4.1) is transformed to z = An (x)z n + An−1 (x)z n−1 + · · · + A1 (x)z + A0 (x), (4.2) Theorem 4.1 For ≤ i ≤ n − 1, we have i−1 An−i = − (−1) j=0 i−j n−j i−j ni−j i−j An−1 An−j + ani−j i−1 i−j (−1) j=0 n−j i−j ni−j i−j an−1 an−j + an−i (4.3) ani−j Theorem 4.2 We have n j=0 (−1)j Aj Ajn−1 An−1 + j j n nAn An n = j=0 (−1)j aj ajn−1 an−1 + j j n nan an 12 4.1.2 Differential invariant under the transformation z = aw Equation (4.2) is transformed to w =a ˜n (x)wn + a ˜n−1 (x)wn−1 + · · · + a ˜1 (x)w + a ˜0 (x), (4.4) By eliminating a, we obtain the following invariants a ˜i Ai i−1 = i−1 , n−1 n−1 ˜n An a ∀i = 2, , n − 1, a ˜n An a ˜1 + = A1 + , n − a ˜ n − A n n 1 a ˜0 a ˜nn−1 = A0 Ann−1 (4.5) Remark 4.3 In order to transform equation (4.2) to (4.4) under z = aw, a must be defined by a ˜n = An an−1 In general, a is in an algebraic extension of the field containing coefficients a ˜n and An 4.1.3 Differential invariant under the transformation y = aw + b Theorem 4.4 For n ≥ 3, two polynomial ordinary differential equations (4.1) and (4.4) are equivalent under the transformation y = aw + b if and only if Ki (a) = Ki (˜ a), ≤ i ≤ n − 2, K1 (a) = K1 (˜ a), K (a) = K (˜ a) (4.6) Corollary 4.5 For n ≥ 3, polynomial differential equation (4.1) is equivalent to the following normal form u = un + Kn−2 (a)un−2 + · · · + K1 (a)u + K0 (a) an−1 under the transformation y = α1 u + β1 with α1n−1 = , β1 = − an nan 4.2 Riccati differential equations Theorem 4.6 For n = 2, two Riccati differential equations (4.1) and (4.4) ˜ (a) = are equivalent under the transformation y = aw + b if and only if K ˜ (˜ K a) 13 Example 4.7 Two Kamke’s differential equations (see [15]) no.1.140 : y = −y − y − , and x x 4x + no.1.165 : y = − y2 + y− 2x − x 2x − x 2x − ˜ (a) = under are equivalent since they have the same differential invariant K 4x − the transformation y = aw + b, where a = ,b=− 2x − x 2x − x Remark 4.8 In [9], Czy˙zycki and collaborators studied the equivalence of the Riccati differential equations under the action of some subgroups of the Lie group of equivalent transformations of Riccati differential equations, including the transformation y = aw + b Proposition 4.9 The Riccati differential equation is equivalent to the autonomous differential equation under the transformation y = aw + b if and ˜ (a) is constant only if the differential invariant K ˜ (a) is non-constant and the Riccati differential equation is defined on If K C(x), then we can use the algorithm by Kovacic (J Kovacic 1986, [17]) to compute algebraic solutions of Riccati differential equation Remark 4.10 Based on the classification of Galois differential groups of second order linear differential equations, F Ulmer et al [33, Corollary 1.7] provided possible degrees of the minimal polynomials of algebraic solutions of the Riccati differential equation w = w2 + r(x) Remark 4.11 Regarding the coefficients of minimal polynomials of algebraic solution, A Zharkov (1995) [35, Theorem 1.1] showed that if the Riccati equation w + w2 = r, for r ∈ Q(x), has an algebraic solution, then there exists a minimal polynomial of that solution whose coefficients are all in a field extension of Q with the degree at most 4.3 Abel’s differential equations Abel’s differential equation of the first type is of the form y = a3 y + a2 y + a1 y + a0 , (4.7) 14 where ∈ K Abel’s differential equation of the second type a3 y + a2 y + a1 y + a0 , y = b1 y + b0 can be transformed to the first type (4.7) by the transformation b1 y + b0 = v Under the transformation y = aw + b, Abel’s equation (4.7) is transformed to w = a ˜3 w + a ˜2 w + a ˜1 w + a ˜0 Corollary 4.12 The system of basic differential invariants of Abel’s equation is a3 a22 − , K1 (a) = a1 + K0 (a) = a3 a3 (3a a − a a a + a a − a a + a ) 3 3 3/2 3a3 (4.8) Corollary 4.13 Two Abel’s differential equations y = a3 y + a2 y + a1 y + a0 and w = a ˜3 w + a ˜2 w + a ˜1 w + a ˜0 are equivalent under the transformation y = aw + b if and only if K1 (a) = K1 (˜ a) and K0 (a) = K0 (˜ a) Example 4.14 Two Abel’s differential equations dt = − (t + x − 1)((t + x)2 − 5(t + x) + 7) dx (4.9) and ds = −(s − 1)3 dx are equivalent under the transformation t = s − x + (4.10) Theorem 4.15 We can transform Abel’s differential equation to the following normal form w = w3 + K1 (a)w + K0 (a) (4.11) Remark 4.16 If both K0 (a) and K2 (a) are constant, then Abel’s differential equation is equivalent to an autonomous differential equation Remark 4.17 If K0 (a) = then equation (4.11) is Bernoulli’s differential equation Theorem 4.18 There are different normal forms of Abel’s differential equations: 15 w = a ˜3 w + a ˜1 w, if K0 (˜ a) = 0; w = a ˜3 w + a ˜1 w + 1, if K0 (˜ a) = Remark 4.19 By using transformations of the form x = φ(t), y(x) = a(t)w(t)+ b(t), P Appell [1] transformed a normal form of Abel’s differential equation w = w3 + J(t) and J(t) is an invariant of the transformation The equivalence of two Abel’s differential equations under the above transformations was studied by E S Cheb-Terrab et al [7] 4.4 Rationally parametrizable first-order algebraic ordinary differential equations Definition 4.20 A first-order algebraic ordinary differential equation F (y, y ) = on C(x) is called rationally parametrizable if the corresponding algebraic curve F (y, y1 ) = is rational Definition 4.21 The associated differential equation of F (y, y ) = 0, with respect to P(t) = (u(t), v(t)), is v(t, x) − ∂u(t,x) dt ∂x = ∂u(t,x) dx ∂t (4.12) The problem of determining degree bound for invariant algebraic curves is known as “Poincar´e’s problem” In [4], Carnicer solved this problem in the non-dicritical case If a degree bound for invariant algebraic curves is obtained, then the differential equation has an algebraic general solution, and one can compute it in the affirmative case using the Prelle-Singer procedure in[28, 20] Theorem 4.22 Let G = ΦM • F Suppose that P(t) = (u(t), v(t)) is a properly rational parametrization of F (y, y1 ) = and let Q(t) = ΦM (P(t)) Then G(w, w1 ) = is rationally parameterized by Q(t) and the associated differential equation of G(w, w ) = with respect to Q(t) is the associated differential equation of F (y, y ) = with respect to the rational parametrization P(t) 16 Theorem 4.23 If two rationally parametrizable AODEs are equivalent, then so are their associated equations Theorem 4.24 Let F (y, y ) = and G(w, w ) = be rationally parametrizable first-order AODEs with the corresponding proper rational parametrizaαt + β ˜ tions P(t) = (u(t), v(t)) and Q(s) = (w(s), z(s)) Suppose that s = γt + δ is an equivalence transformation between the associated differential equations au + b with α, β, γ, δ ∈ C(x), αδ − βγ = If there exists M (u) = such that cu + d ˜ αt + β , then ΦM • F = G ΦM (P(t)) = Q γt + δ Algorithm Input: F (y, y ) = 0, G(w, w ) = are rationally parametrizable Output: Check whether two first-order AODEs F (y, y ) = and G(w, w ) = au + b are equivalent under the Măobius transformation M (u) = cu + d Compute a proper rational parametrization P(t) = (u(t), v(t)) of F (y, y ) = ˜ and a proper rational parametrization Q(s) = (w(s), z(s)) of G(w, w ) = ∂u(t) ∂x ∂u(t) ∂t Compute the differential equation associated with P(t), dt dx = v(t)− ˜ Compute the differential equation associated with Q(s), ds dx = z(s)− If the equation dt dx = ∂u(t) ∂x ∂u(t) ∂t v(t)− is not equivalent to ds dx = ∂w(s) ∂x ∂w(s) ∂s ∂w(s) ∂x ∂w(s) ∂s z(s)− , then F (y, y ) = and G(w, w ) = are not equivalent Find α, β, γ, δ ∈ C(x), αδ − βγ = such that the variable change αt + β s = transfoms an associated differential equation to the other γt + δ associated differential equation ˜ Define Q(t) = Q αt + β γt + δ au + b such that Q(t) = ΦM (P(t)), that is cu + d αt + β αt + β ∂M ∂M w ,z = M (u(t)), (u(t)) + (u(t)) · v(t) γt + δ γt + δ ∂x ∂u Find M (u) = 17 au + b does not exist, then F (y, y ) = and G(w, w ) = are cu + d not equivalent If M (u) = au + b does exist, then F (y, y ) = and G(w, w ) = are cu + d equivalent under the transformation ΦM If M (u) = 4.5 Algebraic general solutions of autonomous rationally parametrizable differential equations Within this section we follow [5] Theorem 4.25 Let F (y, y ) = be a rationally parametrizable algebraic ordinary differential equation of order one which is equivalent to an autonomous equation There exists a proper rational parametrization of F (y, y1 ) = such dt v(t) that its associated differential equation is of the form = ∈ C(t), where dx u (t) u(t), v(t) ∈ C(t) are rational functions of t Theorem 4.26 Let F (y, y ) = be a rationally parametrizable algebraic ordinary differential equation of order one on C(x) and in an autonomous equivalence class Let (u(t), v(t)) be a proper rational parametrization of F (y, y1 ) = u (t) u (t) such that ∈ C(t) If dt is a rational function then the equation v(t) v(t) F (y, y ) = has an algebraic general solution In [29], R H Risch provided an algorithm determining whether the integrals of rational functions are represented in elementary functions In particu (t) ular, by using Risch’s algorithm, we can know whether the integral dt v(t) is a rational function Proposition 4.27 Suppose that (u(t), v(t)) is a proper rational parametrizau (t) tion of the curve F (y, y1 ) = Then, the rationality of dt is indepenv(t) dent of the choice of proper rational parametrization of F (y, y1 ) = Remark 4.28 Recently, the works in [10, Theorem 3.2] and [11, Theorem 3.1] u (t) show that the rationality of dt is a necessary condition for the existence v(t) 18 of an algebraic general solution of the autonomously rationally parametrizable algebraic ordinary differential equation F (y, y ) = Algorithm Input: F (y, y ) = is rationally parametrizable and in an autonomous class under the transformation ΦM Output: Compute an algebraic general solution of F (y, y ) = if there exists Use ΦM to compute a proper rational parametrization (u(t), v(t)) of u (t) ∈ C(t) F (y, y ) = such that v(t) u (t) u (t) dt If dt is not a rational function, then F (y, y ) = v(t) v(t) does not have any algebraic general solution Otherwise, go to step Calculate Let A(t) = B(t) u (t) P (t) dt and = u(t) v(t) Q(t) An algebraic general solution of F (y, y ) = is res (A(t) − B(t)(x + C), yQ(t) − P (t), t) = 19 CONCLUSION In this thesis we have achieved the following main results: Propose several properties of the differential total degree of differential polynomials of order one Establish some properties of the solutions of AODEs which are invariant under the group actions of Măobius transformations Provide a set of invariants of polynomial differential equations Show that the associated differential equation of a rationally parametrizable first-order AODE is an invariant Give a normal form of associated differential equations of rationally parametrizable AODEs of an autonomous class 20 LIST OF THE AUTHOR’S PUBLICATIONS RELATED TO THE THESIS Ngo Lam Xuan Chau, Le Minh Duong, Ha Trong Thi, “Liouville solutions of AODEs of order one in genus zero”, Journal of science, Quy Nhon University, ISSN: 1859-0357, 12(3), 2018: pp 5-12 Ngo Lam Xuan Chau, Ha Trong Thi, Măobius transformations on algebraic ODEs of order one and algebraic general solutions of the autonomous equivalence classes”, Journal of Computational and Applied Mathematics, 380 (2020), 112999 21 REFERENCES [1] P Appell Sur les invariants de quelques ´equations diff´erentielles J Math Pures Appl., 5:361– 423, 1889 [2] J M Aroca, J Cano, R Feng, and X 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Theory Code 9460104 : Reviewer : Prof Dr.Sc Phung Ho Hai Reviewer : Ass Prof Dr Truong Cong Quynh Reviewer : Ass Prof Dr Mai Hoang Bien Supervisors: DR NGO LAM XUAN CHAU DR LE THANH HIEU Binh Dinh... under the supervision of Dr Ngo Lam Xuan Chau and Dr Le Thanh Hieu I hereby declare that the results presented in this thesis are new and original Most of them were published in peer-reviewed journals,... Dat and N L X Chau Liouvillian solutions of algebraic ordinary differential equations of order one in genus zero (submitted), 2021 [11] N T Dat and N L X Chau Rational liouvillian solutions of
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