Morales-Ramis theorems via Malgrange pseudogroup
In this paper the authors prove the following results in algebraic setting for integrability by quadratures. Definition: Let \(E(t,y,y',\dots,y^{(n)})\in C[t,y,y',\dots,y^{(n)}]\) be a differential equation of order \(n\) given by an irreducible polynomial. A Liouvillian solution is a solution \(f\) in a differential extension \(K_N\) of \(K_0=C(t)\) built by successive elementary extensions \(K_{i-1}\subset K_i\), \(1\leq i \leq N\), of the form \(K_i=K_{i-1}(u_i)\) with \(u_i\) algebraic over \(K_{i-1}\) or \(u'_i\in K_{i-1}\) or \(\frac{u'_i}{u_i}\in K_{i-1}\). The equation \(E\) is said to be integrable by quadrature if there is a Liouvillian solution \(f\) with \(\text{transc.deg.}\,C(t,y,y',\dots,y^{(n)})/ C(t) =n \) Theorem: If a rational differential equation is integrable by quadratures then the Galois group of its variational equation of order \(q\) along an algebraic solution is virtually solvable. If \(N\) elementary extensions are needed to build the general solution, then the \(N\)th derived Lie algebra of the Galois group is zero.
scientific article; zbMATH DE number 5689400
scientific article