Abstract
Some algebraic and differential equations are explicitly solvable. What does this mean? If an explicit solution is presented, the question answers itself. However, in most cases, every attempt to solve an equation explicitly is doomed to failure. We are then tempted to prove that certain equations have no explicit solutions. It is now necessary to define exactly what we mean by explicit solutions (otherwise, it is unclear what we are trying to prove).
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Notes
- 1.
If f and g are multivalued functions and ∧ is, say, a binary operation, then f ∧ g is a set of multivalued functions. The class defined by a list \(\{f_{1},\mathop{\ldots },f_{n}\}\) of basic functions and a list \(\{\wedge _{1},\mathop{\ldots },\wedge _{m}\}\) of admissible binary operations is, by definition, the minimal set \(\mathcal{C}\) of functions such that all \(f_{i} \in \mathcal{C}\) and \(f \wedge _{j}g \subseteq \mathcal{C}\) whenever \(f,g \in \mathcal{C}\). An obvious modification can be made to include infinite sets of basic functions and admissible functions, such as unary, ternary, etc., operations.
- 2.
A generalized elementary function over a functional differential field K is, by definition, an element of a generalized elementary extension of K.
- 3.
Since P k is irreducible by our assumption, this simply means that DP k is divisible by P k .
- 4.
We use “polar part of the integral” as a single piece of terminology.
- 5.
It suffices to prove that every function \(\varphi: P \rightarrow \mathbb{Q}\) belongs to J 0(P). Indeed, a function \(\varphi: P \rightarrow \mathbb{Q}\) belongs to J 0(P) if and only if the point \(\sum _{a\in P}(k\varphi (a))a\) has finite order in W, where k is the least common multiple of all the values of \(\varphi\).
References
F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions. Am. J. Math. 101, 42–76 (1979)
M. Bronstein, Integration of elementary functions. J. Symbolic Comput. 9, 117–173 (1990)
M. Bronstein, Symbolic Integration I: Transcendental Functions. Algorithms and Computation in Mathematics, vol. 1, 2nd edn. (Springer, Berlin/New York, 2010)
N.G. Chebotarev, Theory of Algebraic Functions (URSS, Moscow, 2007)
J.H. Davenport, On the Integration of Algebraic Functions (Springer, Berlin/New York, 1981)
A.G. Khovanskii, Fewnomials. Translations of Mathematical Monographs, vol. 88 (American Mathematical Society, Providence, 1991)
A.G. Khovanskii, On algebraic functions integrable in finite terms. Funct. Anal. Appl. 49(1) (2015, to appear)
A.G. Khovanskii, O.A. Gelfond, Real Liouville functions. Funkts. Analiz 14, 52–53 (1980); translation in Funct. Anal. Appl. 14, 122–123 (1980)
E.R. Kolchin, Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations. Ann. Math. 49, 1–42 (1948)
J. Liouville, Sur la détermination des intégrales dont la valeur est algébrique. J. de l’École Poly. 14, 124–148 (1833)
J. Liouville, Mémoire sur l’intégration d’une classe d’équations différentielles du second ordre en quantités finies explicites. Journal de mathématiques, pures et appliquees IV, 423–456 (1839)
V.V. Prasolov, Non-elemetarity of integrals of some elementary functions. Mat. prosveschenie, Ser. III 126–135 (2003)
R.H. Risch, The solution of the problem of integration in finite terms. Bull. Am. Math. Soc. 76, 605–608 (1970)
J. Ritt, Integration in Finite Terms. Liouville’s Theory of Elementary Methods (Columbia University Press, New York, 1948)
M. Rosenlicht, Liouville’s theorem on functions with elementary integrals. Pac. J. Math. 24, 153–161 (1968)
M. Rosenlicht, On Liouville’s theory elementary of functions. Pac. J. Math. 65, 485–492 (1976)
M.F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations. Am. J. Math. 103, 661–682 (1981)
M.F. Singer, Formal solutions of differential equations. J. Symbolic Comput. 10, 59–94 (1990)
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Khovanskii, A. (2014). Construction of Liouvillian Classes of Functions and Liouville’s Theory. In: Topological Galois Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38871-2_1
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