## On algebraic integrals of a differential equation

**Authors:**Malykh M.D., Sevastianov L.A., Ying Y.**Issue:**Vol 27, No 2 (2019)**Pages:**105-123**Section:**Computational modeling and simulation**URL:**http://journals.rudn.ru/miph/article/view/22202**DOI:**http://dx.doi.org/10.22363/2658-4670-2019-27-2-105-123

#### Abstract

We consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as an example. Its solution according to the method of Lagutinski’s determinants and its implementation in the form of a Sagemath package are presented. Necessary conditions for the existence of an integral of contracting derivation are given. A derivation of the ring will be called contracting, if such basis B= {m1, m2, … } exists in which Dmi= cimi+o (mi). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices c1, c2, … there are equal ones. This theorem is convenient for applying to the problem of finding an algebraic integral of Briot-Bouquet equation and differential equations with symbolic parameters. A number of necessary criteria for the existence of an integral are obtained, including those for differential equations of the Briot and Bouquet. New necessary conditions for the existence of a rational integral concerning a fixed singular point are given and realized in Sage.

1. De Beaune problem In the theory of differential equations, it is common from the very beginning to choose a class of functions in which solutions of differential equations are sought so wide that the initial problem has solutions for almost all initial data. In the case of symbolic integration, or finding the solution in finite form, on the contrary, this class is constricted to make it possible in a finite number of operations, first, to find out whether the given differential equation has a general solution in this class, and second, to write out this solution explicitly. The simplest class, which could be expected to possess the above two properties, is the set of algebraic functions. The problem of integrating differential equations in algebraic functions arose as early as the 1630s, when Forimond de Beaune proposed to Descartes several “inverse tangent problems” [1, Pp. 510-518]. We formulate this purely algebraic problem as follows. Problem 1 (de Beaune). Clarify whether a given differential equation (, ) + (, ) = 0, , ∈ [, ], (1) has an integral in the field (, ); in the case of a positive answer, write out this integral. Here is the field of constants, commonly represented by ℚ, ℂ or ℚ[, , … ], where , , … are the parameters that enter the differential equation. There is no reason to consider these cases separately, so we assume that is an infinite field of characteristic zero. The interest to the De Beaune problem sometimes faded away, sometimes arose again. At the turn of the XIX-XX centuries, it was due to successes in proving the nonexistence of algebraic integrals of dynamical systems; among the papers of this period worth particular attention are the Poincaré memoir [2, Pp. 35-95] and a series of articles by M.N. Lagutinski [3, 4]; the biographical data were published by J.-M. Strelcyn [5, 6]. Recently, the classical problem of finding an algebraic integral has again become relevant in connection with the development of algorithms for the symbolic solution of differential equations suitable for implementation in modern computer algebra systems [7, 8]. First of all, it should be noted that popular computer algebra systems cannot efficiently recognize differential equations having algebraic integrals. Example 1. To confirm this statement the following test was used. Let , - be arbitrary polynomials, then = / is an integral of the differential equation ( - ) + ( - ) = 0. Taking randomly and , we get the differential equation (, ) + (, ) = 0, , ∈ ℚ[, ]. An attempt to apply standard methods of solving differential equations to this differential equation in the Maple computer algebra system reduces the differential equation to a quadrature of the form ∫ + = , , ∈ ℚ(, ), occupying many screens, moreover, Maple cannot take the written integrals. 1. D. Malykh, L. A. Sevastianov, Yu. Ying, On algebraic integrals… 107 It is worth noting that for the symbolic solution of differential equations in Maple the package DETools [9] is used. Within the second algorithm of DETools the search for integrating factors in the ring ℂ[, ] is executed. The equation generated in the test has several such integrating factors, namely, and , so that Maple would have to cope with the test. However, the following occurs: § symgen returns two integrating factors, whose ratio yields the rational desired integral, § dsolve ignores the second factor and write out a quadrature which it cannot calculate in elementary functions, although the full implementation of Ostrogradski algorithm would cope with this difficulty. Thus, usually Maple cannot recognize an algebraic integral, however, the user can do it himself, looking at the result of applying the function symgen. Insurmountable difficulties arise when and have common factors. The methods implemented by Maple, first of all, relieve the ordinary differential equation to be solved from common factors. The reduced equation may not have integrating factors in the ring ℂ[, ], and finding factors from ℂ(, ) leads to nonlinear equations for the coefficients and requires completely different computational costs for which the developers of symgen did not go. As a result, e.g., when = (2 + )5( - 6 + 1) + 1, = (138 + 5 + 3 + 2)(2 + )4, symgen finds one factor from ℚ[, ] and nothing else. Despite the antiquity of the de Beaune problem, we do not have an algorithm to solve it in a finite number of operations. The de Beaune problem is equivalent to the problem of integrating a partial differential equation - = 0 in the field (, ); we will further briefly write it as = 0. By the method of uncertain coefficients, we can substitute into the equation = 0 the expression = + ⋯ + 1 + ⋯ + and obtain a system of nonlinear algebraic equations for finding the coefficient , , , … . The solvability of this system can be determined in a finite number of steps and in a purely algebraic way. Therefore, in a finite number of operations one can find out whether a given differential equation has rational integrals whose degree does not exceed a given number . The problem of finding the upper bound for the degree of the sought integral was noted by Descartes, and in some cases was resolved by Poincaré [2], pp. 35-95. The idea of the Poincaré method is as follows. If a differential equation admits a rational integral, then its integral curves form a linear sheaf of algebraic curves of some order , this immediately follows from a comparison of the Cauchy theorem from the analytic theory of differential equations [10] and Bertini’s theorem from the theory of algebraic curves [11]. Two arbitrary curves of the sheaf intersect at 2 fixed points. On the other hand, according 108 DCM&ACS. 2019, 27 (2) 105-123 to the Cauchy theorem, these curves can intersect only at those points at which the polynomials and vanish simultaneously; in the analytical theory of differential equations, such singular points are called fixed points. If the orders of the curves (, ) = 0 and (, ) = 0 do not exceed , then ⩽ . However, it is impossible to bring this idea to a rigorous statement: among the intersection points of the integral curves there may be multiple and infinitely distant ones, as well as at fixed singular points of the differential equation, the solutions may have various kinds of “degeneracies”. That is why M.N. Lagutinski carefully notes that the “French scientist in the work just referred deduces a number of equalities and inequalities that in some cases achieve the goal of indicating the upper bound of the order ” [3, P. 181]. Taking into account that “the difficulties of this way for solving this problem have stopped even H. Poincaré” [3], it is not hard to understand why in all modern implementations of algorithms for finding integrals, the order of the integral is assumed to be given [12]. The de Beaune problem, in which a bound for the orders of considered integrals is given, will be referred to as a bounded problem. Problem 2 (The bounded de Beaune problem). Clarify whether a given differential equation (, ) + (, ) = 0, , ∈ [, ], (2) admits an integral in the field (, ) whose order does not exceed a given number , and in case of positive answer, write out this integral. Practically the described solution of a system of nonlinear algebraic equations requires considerable computation resources even at = 3. Therefore, the authors of algorithms for solving this problem try to avoid the solution of nonlinear systems. Among the implemented algorithms, worth special attention are the Lagutinski’s method of determinants and the method proposed by Jacques-Arthur Weil in 1985 based on power series expansion [12]. 2. The bounded de Beaune problem and Lagutinski’s method of determinants Lagutinski’s method allows searching for particular and general integrals of ring derivations of sufficiently general form. An up-to-date presentation of this method for the case of the ℂ[, ] ring is given in [13, 14], and the general case is considered in [15]. For convenience of reference we present here a brief description of the method. Let be a ring with derivation and field of constants . Consider to be an arbitrary field of characteristic zero and ℚ ∈ . Let us call a general integral of this derivation a pair of elements 1, 2 linearly independent over the field , satisfying the equality 12 = 21. (3) If the ring is integral, then the derivation is naturally continued on its field of quotients, and the fraction 1/2 satisfies the equation (1/2) = 0. M. D. Malykh, L. A. Sevastianov, Yu. Ying, On algebraic integrals… 109 We will deal with rings where a basis can be introduced in the following sense. Definition 1. A countable ordered set of elements of a ring will be called a basis of the ring if 1. any element of the ring can be presented as a linear combination of a finite number of elements of the set with constant coefficients; 2. a product of any two elements of the set belongs to , and follows strictly after both efficients, i.e., = and is strictly greater than and . Let us introduce the ordering relationship in the basis, i.e., the inequality < means that < and assume that the notation = () means that the representation of the element of the ring in the form of a linear combination of basis elements contains the basis elements whose numbers are strictly larger than . If = + (), ≠ 0, then the addend will be called the lowest term in . In contrast to the common agreement, we call the number of the greatest basis term entering the decomposition of an element in the basis an order of this element. Example 2. In the ring = ℚ[, ] a system of various monomials may be taken to be a basis by accepting the glex-ordering: 1, , , 2, , 2, 3, 2, 2, 3, … Below this basis will be referred to as glex-basis. In this case, for example, 2 + + 33 = 2 + (2), and the order of this element equals 10. The calculations of integrals is closely related to Lagutinski’s determinants. Definition 2. Compose an infinite matrix with the first row 1, 2, … , the second row being the first derivative of the first one, 1, 2, … , the third row being the second derivative of the first one, 21, 22, … , and so on to infinity. A determinant of the corner minor of the -th order of this matrix, i.e., ⎛ ⎜ ⎜ det ⎜ ⎜ 1 2 … ⎞ ⎟ ⎟ 1 2 … ⎟ ⋮ ⋮ ⋱ ⋮ ⎟ (4) ⎝-11 -12 … , -1⎠ 110 DCM&ACS. 2019, 27 (2) 105-123 will be denoted by Δ and called Lagutinski’s determinant of the -th order. The following theorem provides a complete solution of the bounded de Beaune problem. Theorem 1 (by M. N. Lagutinski). Let be a ring of polynomials. 1. A general integral exists then and only then, when all Lagutinski’s determinants of sufficiently high order are equal to zero. 2. A general integral of the order exists then and only then, when Δ = 0; in this case the integral can be calculated as a ratio of the corresponding minors of this determinant. The proof of Lagutinski’s theorem and the rule of choosing minors to construct integrals is given in [15]. Remark 1. From this theorem, in particular, it follows that finding a rational integral does not require the field extension. If and belong to ℚ[, ] and there is an integral in ℂ(, ), then applying this theorem at = ℂ, we see that for a certain Δ = 0. The calculation of Lagutinski’s determinants does not lead beyond the field ℚ. Therefor, applying this theorem at = ℚ, we arrive at the existence of an integral in the field ℚ(, ). For this reason, below we mean the integral of an equation with integer coefficients to be an element of ℚ(, ). Lagutinski’s method agrees well with the concept of operating with rings, accepted in Sage [16]. We have written a package Lagutinski [17] in Sage, which allows calculation of Lagutinski’s determinants and integrals in this environment. The package was presented in 2016 at a number of conferences on computer algebra [18-20]. Here we restrict ourselves to one example illustrating the application of this package. In more detail the technique of its application is described in [21]. Example 3. Let the Bernoulli differential equation be given, ( + 1) - (2 + + 2) = 0, which for certain possesses an algebraic integral. Let us find it using Lagutinski’s method. For this purpose we specify in a usual manner the corresponding differential ring and its basis: sage: R.

### Mikhail D Malykh

Peoples’ Friendship University of Russia
**Author for correspondence.**

Email: malykh-md@rudn.ru

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics

### Leonid A Sevastianov

Peoples’ Friendship University of Russia
Email: sevastianov-la@rudn.ru

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

professor, Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics

### Yu Ying

Peoples’ Friendship University of Russia
Email: yingy6165@gmail.com

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 3, Kaiyuan Road, Kaili, 556011, China

postgraduate student of Department of Applied Probability and Informatics; assistant professor of Department of Algebra and Geometry

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