Radical Solutions of First Order Autonomous Algebraic Ordinary Differential Equations

Transcription

1 Radical Solutions of First Order Autonomous Algeraic Ordinary Differential Equations Georg Grasegger Johannes Kepler University Linz Doctoral Program Computational Mathematics 4040 Linz, Austria ABSTRACT We present a procedure for solving autonomous algeraic ordinary differential equations AODEs of first order. This method covers the known case of rational solutions and depends crucially on the use of radical parametrizations for algeraic curves. We can prove that certain classes of AODEs permit a radical solution, which can e determined algorithmically. However, this approach is not limited to rational and radical solutions of AODEs. Categories and Suject Descriptors G..7 [Mathematics of computing]: Ordinary differential equations; I.. [Computing methodologies]: Symolic and algeraic manipulation General Terms Algorithms, Theory Keywords Algeraic ordinary differential equations, algeraic curves, radical parametrizations. INTRODUCTION We consider autonomous first order algeraic ordinary differential equations F y, y = 0 and use parametrizations of curves for solving them. Huert [8] already studied solutions of AODEs of the form F x, y, y = 0. She gives a method for finding a asis of the general solution of the equation y computing a Gröner asis of the prime differential ideal of the general component. The solutions, however, are given implicitly. Later Feng and Gao [, 3] started using parametrizations for solving first order autonomous AODEs, i. e. F y, y = 0. They provide an algorithm to actually find Supported y the Austrian Science Fund FWF: W4- N5, project DK Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distriuted for profit or commercial advantage and that copies ear this notice and the full citation on the first page. Copyrights for components of this work owned y others than ACM must e honored. Astracting with credit is permitted. To copy otherwise, or repulish, to post on servers or to redistriute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org. ISSAC 4, July 5, 04, Koe, Japan. Copyright 04 ACM /4/07...$ all rational solutions of such AODEs with coefficients in Q y using rational parametrizations of the algeraic curve F y, z = 0. The key fact they are proving is that any rational solution of the AODE gives a proper parametrization of the corresponding algeraic curve. On the other hand, if a proper parametrization of the algeraic curve fulfills some requirements, Feng and Gao can generate a rational solution of the AODE. From the rational solution it is then possile to create a rational general solution y shifting the variale y a constant. Finally Feng and Gao derive an algorithm to compute a rational solution of an AODE, if it exists. By using this approach we can take advantage of the well known theory of algeraic curves and rational parametrizations see for instance [, 3]. Recently Ngô and Winkler [, 4, 3] worked on generalizing the algorithm of Feng and Gao. They considered nonautonomous AODEs F x, y, y = 0. Instead of algeraic curves as in the autonomous case, algeraic surfaces play a role there. For an algorithm to find rational parametrizations of surfaces see [7]. Using a rational parametrization of the surface Ngô and Winkler derive a special kind of system of differential equations. There exist solution methods for these so called associated systems. From a rational solution of such a system they find a rational general solution of the original differential equation in the generic case. Aroca, Cano, Feng and Gao give in [] a necessary and sufficient condition for an autonomous AODE to have an algeraic solution. They also provide a polynomial time algorithm to find such a solution if it exists. This solution, however, is implicit whereas here we try a different approach to find explicit solutions. Furthermore Huang, Ngô and Winkler continue their work and consider higher order equations. A first result can e found in [7]. A generalization of Ngô and Winkler [4] to trivariate systems of ODEs can e found in [5, 6]. Ngô, Sendra and Winkler [, 0] also consider classification of AODEs, i. e. they look for transformations of AODEs which keep the associated system invariant. In this work we stick to the case of first order autonomous AODEs ut try to extend the results to radical solutions using radical parametrizations see Section 3.. We do so y investigating a procedure which can e specified to the rational case see Section 3.. In contrast to the rational case we will not give a complete algorithm ut rather a procedure which in the favorale case yields a result. If a result is computed then it will e a solution of the AODE. On the other hand the procedure might fail and then we cannot draw any conclusion on the solvaility of the AODE.

2 Having computed a single non-trivial solution yx we can get a general solution yx+c for a constant c as shown in []. In Section 3.3 we give examples of non-radical solutions that can e found y the given procedure. Finally in Section 3.4 we look for advantages of the procedure and compare it to existing algorithms.. PRELIMINARIES In the following we descrie the necessary notations and definitions from differential algera and algeraic geometry. Throughout the paper let K e a field of characteristic zero and K its algeraic closure.. Solutions of AODEs Consider the field of rational functions Kx. We will write for the usual derivative d. Then Kx is a differential field. We call Kx{y} the ring of differential poly- dx nomials. The elements of this ring are polynomials in y and its derivatives, i. e. Kx{y} = Kx[y, y, y,...]. An algeraic ordinary differential equation AODE is of the form F x, y, y,..., y n = 0 where F Kx{y} and F is also a polynomial in x. The AODE is called autonomous if F K{y}, i. e. if the coefficients of F do not depend on the variale of differentiation x. For a given AODE we are interested in deciding whether it has rational or radical solutions and, in the affirmative case, determining all of them. Hence, we look for general solutions which are defined as follows. Let Σ e a prime differential ideal in Kx{y}. Then we call η a generic zero of Σ if for any differential polynomial P we have P η = 0 P Σ. Such an η exists in a suitale extension field. Let F e an irreducile differential polynomial of order n. Then {F }, the radical differential ideal generated y F, can e decomposed into two parts. There is one component F where the separant also vanishes. This part represents y n the singular solutions. The component we are interested in is the one where the separant does not vanish. It is a prime differential ideal Σ F := {F } : F and represents the general component see for instance Ritt [6]. A generic zero of y n Σ F is called a general solution of F = 0. We say it is a rational general solution if it is of the form y = a kx k +...+a x+a 0 mx m x+ 0, where the a i and i are constants in some field extension of K. Here we consider only first order AODEs. For solving a differential equation F y, y = 0 we will look at the corresponding curve F y, z = 0 where we replace the derivative of y y a transcendental variale z.. Algeraic Curves An algeraic curve C is a one-dimensional algeraic variety in the affine plane A over a field K, i. e. a zero set of a square-free ivariate non-constant polynomial f K[x, y], C = { a, A fa, = 0 }. We call the polynomial f the defining polynomial. An important aspect of algeraic curves is their parametrizaility. Consider an irreducile plane algeraic curve defined y an irreducile polynomial f. A tuple of rational functions Pt = rt, st is called a rational parametrization of the curve if frt, st = 0 and not oth rt and st are constant. A parametrization can e considered as a map P : A C from the affine line A onto the curve C. By ause of notation we also call this map a parametrization. Later we will see other kinds of parametrizations. We call a parametrization P proper if it is a irational map or in other words if for almost every point x, y on the curve we find exactly one t such that Pt = x, y. In the procedure we will assume that the input is a radical parametrization. The research area of radical parametrizations is rather new. Sendra and Sevilla [9] recently pulished a paper on parametrizations of curves using radical expressions. In this paper Sendra and Sevilla define the notion of radical parametrization and they provide algorithms to find such parametrizations in certain cases which include ut are not restricted to curves of genus less or equal 4. Every rational parametrization will e a radical one ut oviously not the other way round. Further considerations of radical parametrizations can e found in Schicho and Sevilla [8] and Harrison [4]. There is also a paper on radical parametrization of surfaces y Sendra and Sevilla [0]. Nevertheless, for the eginning we will restrict to the case of first order autonomous equations and hence to algeraic curves. Definition. Let K e an algeraically closed field of characteristic zero. A field extension K L is called a radical field extension iff L is the splitting field of a polynomial of the form x k a K[x], where k is a positive integer and a 0. A tower of radical field extensions of K is a finite sequence of fields K = K 0 K K... K m such that for all i {,..., m}, the extension K i K i is radical. A field E is a radical extension field of K iff there is a tower of radical field extensions of K with E as its last element. A polynomial hx K[x] is solvale y radicals over K iff there is a radical extension field of K containing the splitting field of h. Let now C e an affine plane curve over K defined y an irreducile polynomial fx, y. According to [9], C is parametrizale y radicals iff there is a radical extension field E of Kt and a pair rt, st E \ K such that frt, st = 0. Then the pair rt, st is called a radical parametrization of the curve C. We call a function fx over K a radical function if there is a radical extension field of Kx containing fx. Hence, a radical solution of an AODE is a solution that is a radical function. A radical general solution is a general solution which is radical. Computing radical parametrizations as in [9] goes ack to solving algeraic equations of degree less or equal four. Depending on the degree we might therefore get more than one solution to such an equation. Each solution yields one ranch of a parametrization. Therefore, we use the notation a n for any n-th root of a. In fact we do not need to restrict to rational or radical parametrizations. More generally a parametrization of f is a generic zero of the the prime ideal generated y f in the sense of van der Waerden []. 3. A PROCEDURE FOR SOLVING FIRST ORDER AUTONOMOUS AODES Let F y, y = 0 e an autonomous AODE. We consider the corresponding algeraic curve F y, z = 0. Then oviously P yt := yt, y t for a non-trivial, i. e. non-

3 constant solution y of the AODE is a parametrization of F not necessarily rational or radical. For an aritrary parametrization Pt = rt, st we define A Pt := st. If it is clear which parametrization r t is considered, we might also write A. Suppose we are given any parametrization P of F and we want to find a solution of the AODE. We assume the parametrization is of the form P gt = rt, st = ygt, y gt where g and y are unknown. This assumption is motivated y the fact that in the rational case each proper parametrization can e otained from any other proper one y a linear rational transformation of the variale. If we can find g and especially its inverse function we also find y. For our given P g we can compute A Pg and furthermore we have A Pg t = y gt d ygt = y gt g ty gt = g t. dt Hence, y reformulation we get an expression for the unknown g : g t = A Pg t. Using integration and inverse functions the procedure continues as follows gt = g tdt = A Pg t dt, yx = rg x. Kamke [9] already mentions such a procedure where he restricts to continuously differential functions r and s which satisfy F rt, st = 0. However, he does not mention where to get these functions from. In general g is not a ijective function. Hence, when we talk aout an inverse function we actually mean one ranch of a multivalued inverse. Each ranch inverse will give us a solution to the differential equation. We might add any constant c to the solution of the indefinite integral. Assume gt is a solution of the integral and g its inverse. Then also ḡt = gt + c is a solution and ḡ t = g t c. We know that if yx is a solution of the AODE, so is yx + c. Hence, we may postpone the introduction of c to the end of the procedure. We summarize the procedure for the case of radical parametrizations and solutions Procedure. Input: an autonomous AODE defined y F y, y = 0 and a radical parametrization Pt = rt, st of the corresponding curve. Output: a general solution y of F or fail.. Compute A Pt = st r t.. Compute gt = A P t dt. 3. Compute g. 4. If g is not radical return fail, else return rg x + c. The procedure finds a solution if we can compute the integral and the inverse function. On the other hand it does not give us any clue on the existence of a solution in case either part does not work. Neither do we know whether we found all solutions. 3. Rational solutions Feng and Gao [] found an algorithm for computing all rational general solutions of an autonomous first order AODE. They use the fact that yx, y x is a proper rational parametrization. The main part of their algorithm says that there is a rational general solution if and only if for any proper rational parametrization Pt = rt, st we have that A Pt = q Q or A Pt = at with a, Q. The solutions therefore are rqx + c or r respectively. ax+c We will show now that the algorithm accords with our procedure. Assume we are given an AODE with a proper parametrization Pt = rt, st. Assume further that A Pt = q Q or A Pt = at. Then we get from the procedure A Pt = q, A Pt = at, g t = q, g t = at, gt = t q + c, gt = at + c, g t = qt c, g t = + at c. at c We see that rg t is exactly what Feng and Gao found aside from the sign of c. As mentioned aove, our procedure does not give an answer to whether the AODE has a rational solution in case A is not of this special form. It might, however, find non rational solutions for some AODEs. Nevertheless, Feng and Gao [] already proved that there is a rational general solution if and only if A is of the special form mentioned aove and all rational general solutions can e found y the algorithm. 3. Radical solutions Now we extend our set of possile parametrizations and also the set, in which we are looking for solutions, to functions including radical expressions. We will see that the procedure yields information aout solvaility in some cases. Theorem. Let Pt = rt, st e a radical parametrization of the curve F y, z = 0. Assume A Pt = a+t n for some n Q \ {}, a 0. Then rhx, with hx = + n ax + c n, is a radical general solution of the AODE F y, y = 0. Proof. From the procedure we get g t = A P t = a + t, n gt = g + t n tdt = a n, g t = + n at n. Then yx = rg x is a solution of F. Let hx = g x + c for some constant c. Then rhx is a general solution of F. The algorithm of Feng and Gao is therefore a special case of this one with n = 0 or n = using rational parametrizations. In exactly these two cases g is a rational function. Furthermore, Feng and Gao [] showed that all rational solutions can e found like this, assuming the usage of a rational parametrization. The existence of a rational parametrization is of course necessary to find a rational solution.

4 However, in the procedure we might use a radical parametrization of the same curve which is not rational and we can still find a rational solution. In Theorem n = is excluded ecause in this case the function g contains a logarithm and its inverse an exponential term. Example. The equation y 5 y = 0 gives rise to the radical parametrization, with corresponding At = t t 5/ t. We can compute gt = t3/ and g t = 3 /3 t / Hence, /3 is a solution of the AODE. x+c /3 As a corollary of Theorem we get the following statement for AODEs where the parametrization yields another form of A. Corollary. Let F y, y = 0 e an autonomous AODE. Assume we have a radical parametrization Pt = rt, st of the corresponding curve F y, z = 0 and assume At = a+t k n kt k with k Q. Then F has a radical solution. Proof. By transforming the parametrization y ft = t /k to the radical parametrization Pt = rft, sft we compute A Pt = t sft Aft = rft f t = a + t k k n kt k k t k k k = a + t n, which can e solved y Theorem. = a + ftk n kft k f t In the rational situation there were exactly two possile cases for A in order to guarantee a rational solution. Here, in contrast, there are more possile forms for A. In the following we will see another rather simple form of A which might occur. In this case we do not know immediately whether or not the procedure will lead to a solution. Theorem. Let Pt = rt, st e a radical parametrization of the curve F y, z = 0. Assume At = atn for +t m some a, Q\{0} and m, n Q with m n and n. Then the AODE F y, y = 0 has a radical solution if the equation m n + h n n h m n+ + n m n + at = 0 has a non-zero radical solution for h = ht. solution of the AODE is then rhx + c. Proof. The procedure yields A general g t = A = + tm Pt at, n gt = g tdt = a t n n + t m. + m n The inverse of g can e found y solving the equation a ht n n + htm = t + m n for ht. By a reformulation and the assumptions for m and n this is equivalent to. In case we have n = or m = n the integral is not radical. Example. For the AODE y 5 y + y 8 y = 0 we compute the radical parametrization Pt =, t 3 with cor- t t 8 responding At = t5. Then equation has a solution, +t 8 e. g. t + 4t /4. Hence, we get the solution of the AODE: x + c /4. 4x + c It remains to show when is solvale y radicals i. e. when gt in the proof of Theorem has an inverse which is expressile y radicals. The following theorem due to Ritt [5] will help us to do so. Theorem 3. A polynomial g has an inverse expressile y radicals if and only if it can e decomposed into linear polynomials, power polynomials x n for n N, Cheyshev polynomials and degree 4 polynomials. Certainly also polynomials of degree and 3 are invertile y radicals ut it can e shown, that Theorem 3 applies to them. We will show now that a certain polynomial is not decomposale into non-linear factors. Theorem 4. Let gt = C t α + C t β K[t] where K is a field of characteristic zero, C, C K \ {0}, α, β N, gcdα, β = and β > α > 0 and β > 4. Assume g = fh for some polynomials f and h. Then deg f = or deg h =. Proof. Assume gt = fht with f = n i=0 aixi and h = m k=0 kx k and a n 0, m 0, m, n >. In case 0 0 it follows that gt = f ht where ft = f 0 + t and ht = ht 0. Hence, without loss of generality we can assume that 0 = 0 and therefore also a 0 = 0. We denote the coefficient of a polynomial g of order k y coef k g. Let now τ {,..., m} such that τ 0 and l = 0 for all l {,..., τ }. Similarly let π {,..., n} such that a π 0 and a l = 0 for all l {,..., π }. This implies that coef l g = 0 for all l {,..., τπ } and coef τπg = a π π τ 0. Hence, α = τπ. Assume now that π < n. Then α = τπ < mn + l and therefore 0 = coef mn +l g = coef mn +l = na n n m l + ε E m n a n jx j j=τ n ε l+,..., ε m a n m j=l+ for all l {τ,..., m } where ε = ε l+,..., ε m and { } m m E = ε ε k = n, kε k = mn + l. k=l+ k=l+ This yields, that 0 = coef mn g = na n n m m, hence, m = 0. By induction it follows that l = 0 for all l {τ,..., m } which contradicts τ 0. Therefore, τ = m or π = n. But then we have m α and m β or n α and n β which contradicts gcdα, β = since m, n. ε j j

5 Let us now consider the function gt = a t n n + t m + m n from the proof of Theorem. In general g is not a polynomial. Our aim is to find a radical function h with a radical inverse such that ḡ = gh is a polynomial. Then g has a radical inverse if and only if ḡ has a radical inverse. Assume that n = z d and m n+ = z d with z, z Z, d, d N such that gcdz, d = gcdz, d =. Then gt = ḡht where ht = t d d and ḡt = t n a n + t m n + m n with exponents n = nd d, m = m n+d d and d = d d gcdz d, z d. Hence, m, n are integers with gcd m, n =. The function h has an inverse expressile y radicals. If m n +, n are positive integers, then also m, n are positive. On the other hand if n m, n N we get a polynomial y a decomposition with ft = t. If not oth n and m are positive and not oth are negative ut m + n 4, computing the inverse function of ḡ is the same as solving an equation of degree less or equal 4, which can e done y radicals. We summarize this discussion using Theorem 3 and 4 as follows. Corollary. The function g from the proof of Theorem, gt = a t n n + t m, + m n has an inverse expressile y radicals in the following cases where we use the notation from aove: = 0, m, n N and max m, n 4, m, n N and max m, n 4, m, n N and m + n 4, m, n N and m + n 4. It has no inverse expressile y radicals in the cases m, n N and max m, n > 4, m, n N and max m, n > 4. Proof. In the cases where m and n have the same sign and m + n > 4 the function ḡ as discussed aove fulfills the requirements of Theorem 4. Hence, if ḡ = f f either f or f is of degree one. It is not difficult to show, that the other one can neither e a power polynomial nor a Cheyshev polynomial. Hence, y the Theorem of Ritt ḡ has no radical inverse. The case where = 0 is ovious. In all the other cases mentioned in the Theorem and not discussed so far we end up in solving an algeraic equation of degree less or equal four and hence, there is a radical inverse. Hence, in some cases we are ale to decide the solvaility of an AODE with properties as in Theorem. Nevertheless, the procedure is not complete, since even Corollary does not cover all possile cases for m and n. d 3.3 Non-Radical solutions So far we were looking for rational and radical solutions of AODEs. However, the procedure is not restricted to the radical case ut might also solve some AODEs with nonradical solutions as we can see in the following examples where trigonometric and exponential solutions are found. Example 3. Consider the equation y 3 + y + y = 0. The corresponding curve has the parametrization P t = t, t t. We get At = + t and hence, gt = arctant. The inverse function is g t = tan t and thus, yx = tan x+c is a solution. Example 4. Consider the AODE y + y + yy + y = 0. We get the rational parametrization, t. +t +t With At = t + t we compute gt = logt + log + t and hence g t =, which leads to the +e t/ solution e x+c + e x+c/. These examples show that we can find non-radical solutions even with rational parametrizations. 3.4 Comparison In many ooks on differential equations we can find a method for transforming an autonomous ODE of any order F y, y,..., y n = 0 to an equation of lower order y sustituting uy = y see for instance [4, 9]. For the case of first order ODEs this method yields a solution. It turns out that this method is somehow related to our procedure. The method does the following: Sustitute uy = y, Solve F y, uy = 0 for uy, Solve dy = x for y. uy These computations are a special case of our general procedure where a specific form of parametrization is used, i. e. Pt = t, st. We will now give some arguments concerning the possiilities and enefits of the general procedure. Since in the procedure any radical parametrization can e used we might take advantage of picking a good one as we will see in the following example. Example 5. We consider the AODE y yy 7 and find a parametrization of the form t, st: t3 + 8 /3 588t t, t /6 Neither Mathematica 8 nor Maple 6 can solve the corresponding integral explicitly and hence, the procedure stops. Furthermore, neither of them is capale of solving the differential equation in explicit form y the uilt in functions for solving ODEs. Nevertheless, we can input another parametrization to our procedure. An ovious one to try next is 7 + t6 rt, st =, t. 49t

6 It turns out that here we get gt = 4t3. Its inverse t 3 47 can e computed g t = /3 47t t. Applying g x + c to rt we get the solution 7 47x + c x + c 64 yx = 47x + c 7 / x + c 49 4 The procedure might find a radical solution of an AODE y using a rational parametrization as we have seen in Example and 5. As long as we are looking for rational solutions only, the corresponding curve has to have genus zero. Now we can also solve some examples where the genus of the corresponding curve is higher than zero and hence there is no rational parametrization. The AODE in Example 6 elow corresponds to a curve with genus. Example 6. Consider the AODE y 3 4y 5 + 4y 7 y 8y y + 8y 4 y + 8yy = 0. We compute a parametrization and get t, 4 + 4t + t 4 t 4t 4t 4 t 6 6t 4 + 6t 8 + 8t 0 + t as one of the ranches. The procedure yields t 4 + 4t + t 4 At = 4t 4t 4 t 6 6t 4 + 6t 8 + 8t 0 + t, gt = t4 + t 6 + t 4 + t 4 + 4t + t 4, 4t + t 4 g + t t =, + t + c + x yx =. + c + x Again Mathematica 8 cannot compute a solution in reasonale time and Maple 6 only computes constant and implicit ones. 4. CONCLUSION We have introduced a procedure for solving autonomous first order ordinary differential equations. The procedure works in a given class of expressions if we can compute a certain integral and an inverse function in this class. In case of looking for rational solutions it does exactly what was known efore. Furthermore, we have found some cases in which we find radical solutions. However, these cases are not yet complete and hence, this part is suject to further investigation. In case the procedure works, we have one or more solutions of the AODE. So far we do not know, whether we have all. Neither do we know anything aout solvaility of the AODE if the procedure does not work. We have seen that the choice of parametrization makes a difference. The influence of the choice of parametrization on the solvaility of the integration and inversion prolem is also a topic for further investigation. 5. ACKNOWLEDGMENTS The author cordially thanks Franz Winkler and J. Rafael Sendra for their valuale support during this work. 6. REFERENCES [] J. M. Aroca, J. Cano, R. Feng, and X.-S. Gao. Algeraic General Solutions of Algeraic Ordinary Differential Equations. In M. Kauers, editor, Proceedings of the 005 international symposium on symolic and algeraic computation ISSAC, Beijing, China, pages 9 36, New York, 005. ACM Press. [] R. Feng and X.-S. Gao. Rational General Solutions of Algeraic Ordinary Differential Equations. In J. Gutierrez, editor, Proceedings of the 004 international symposium on symolic and algeraic computation ISSAC, pages 55 6, New York, 004. ACM Press. [3] R. Feng and X.-S. Gao. A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs. Journal of Symolic Computation, 47:739 76, 006. [4] M. Harrison. Explicit solution y radicals, gonal maps and plane models of algeraic curves of genus 5 or 6. Journal of Symolic Computation, 5:3, 03. [5] Y. Huang, L. X. C. Ngô, and F. Winkler. Rational General Solutions of Trivariate Rational Systems of Autonomous ODEs. In Proceedings of the Fourth International Conference on Mathematical Aspects of Computer and Information Sciences MACIS 0, pages 93 00, 0. [6] Y. Huang, L. X. C. Ngô, and F. Winkler. Rational General Solutions of Trivariate Rational Differential Systems. Mathematics in Computer Science, 64:36 374, 0. [7] Y. Huang, L. X. C. Ngô, and F. Winkler. Rational General Solutions of Higher Order Algeraic ODEs. Journal of Systems Science and Complexity, 6:6 80, 03. [8] E. Huert. The General Solution of an Ordinary Differential Equation. In Y. Lakshman, editor, Proceedings of the 996 international symposium on symolic and algeraic computation ISSAC, pages 89 95, New York, 996. ACM Press. [9] E. Kamke. Differentialgleichungen: Lösungsmethoden und Lösungen. B. G. Teuner, Stuttgart, 997. [0] L. X. C. Ngô, J. R. Sendra, and F. Winkler. Birational Transformations on Algeraic Ordinary Differential Equations. Technical Report 8, RISC Report Series, Johannes Kepler University Linz, Austria, 0. [] L. X. C. Ngô, J. R. Sendra, and F. Winkler. Classification of algeraic ODEs with respect to rational solvaility. In Computational Algeraic and Analytic Geometry, volume 57 of Contemporary Mathematics, pages American Mathematical Society, Providence, RI, 0. [] L. X. C. Ngô and F. Winkler. Rational general solutions of first order non-autonomous parametrizale ODEs. Journal of Symolic Computation, 45:46 44, 00. [3] L. X. C. Ngô and F. Winkler. Rational general solutions of parametrizale AODEs. Pulicationes Mathematicae Derecen, 793 4: , 0. [4] L. X. C. Ngô and F. Winkler. Rational general solutions of planar rational systems of autonomous odes. Journal of Symolic Computation,

7 460:73 86, 0. [5] J. F. Ritt. On algeraic functions, which can e expressed in terms of radicals. Transactions of the American Mathematical Society, 4: 30, 94. [6] J. F. Ritt. Differential algera, volume 33 of Colloquium Pulications. American Mathematical Society, New York, 950. [7] J. Schicho. Rational Parametrization of Surfaces. Journal of Symolic Computation, 6: 9, 998. [8] J. Schicho and D. Sevilla. Effective radical parametrization of trigonal curves. In Computational Algeraic and Analytic Geometry, volume 57 of Contemporary Mathematics, pages 3. American Mathematical Society, Providence, RI, 0. [9] J. R. Sendra and D. Sevilla. Radical parametrizations of algeraic curves y adjoint curves. Journal of Symolic Computation, 469: , 0. [0] J. R. Sendra and D. Sevilla. First steps towards radical parametrization of algeraic surfaces. Computer Aided Geometric Design, 304: , 03. [] J. R. Sendra, F. Winkler, and S. Pérez-Díaz. Rational Algeraic Curves, A Computer Algera Approach, volume of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin Heidelerg, 008. [] B. L. van der Waerden. Algera, volume II. Springer-Verlag, New York, 99. [3] R. J. Walker. Algeraic curves. Reprint of the st ed. 950 y Princeton University Press. Springer-Verlag, New York - Heidelerg - Berlin, 978. [4] D. Zwillinger. Handook of Differential Equations. Academic Press, San Diego, CA, third edition, 998.