Consider a linear differential equation L(f) = 0 with coefficients in \(\overline{\mathbb {Q}}[x,\sqrt {S(x)}]\) where \(S\in \overline{\mathbb {Q}}[x]\) square free. We present an algorithm to compute hyperexponential solutions of L, i.e. solutions L(f) = 0 such that \(f^{\prime }/f \in \overline{\mathbb {Q}}(x,\sqrt {S(x)})\). We then present a p-curvature criterion to discard impossible combinations at a lower cost.

Consider a third order linear differential equation L(f) = 0, where <?TeX $L\in \mathbb {Q}(z)[\partial _z]$?> Math 1 . We design an algorithm computing the Liouvillian solutions of L(f) = 0. The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function 3F2. The computation time is practical for reasonable size operators.

Consider a hyperelliptic integral , , with . When S is of degree ≤ 4, such integral can be calculated in terms of elementary functions and elliptic integrals of three kinds . When S is of higher degree, it is typically non elementary, but it is sometimes possible to obtain an expression of I using also elliptic integrals when the Jacobian of y2 = S(x) has elliptic factors. We present an algorithm searching for elliptic factors and a modular criterion for their existence. Then, we present an algorithm for computing an expression of I using elliptic integrals, which always succeed in the completely decomposable Jacobian case.

Consider a superelliptic integral $I=\int P/(Q S^1/k ) dx$ with $\mathbbK =\mathbbQ (ξ)$, ξ a primitive kth root of unity, $P,Q,S\in\mathbbK [x]$ and S has simple roots and degree coprime with k. Note d the maximum of the degree of $P,Q,S$, h the logarithmic height of the coefficients and g the genus of $y^k-S(x)$. We present an algorithm which solves the elementary integration problem of I generically in $O((kd)^ømega+2g+1 h^g+1 )$ operations.

Let H be a hyperexponential function in n variables x=(x1,…,xn) with coefficients in a field ℚ , [ℚ :ℚ ] < ∞, and ω a rational differential 1-form. Assume that Hømega is closed and H transcendental. We prove using Schanuel conjecture that there exist a univariate function f and multivariate rational functions F,R such that ∫ Hω= f(F(x))+H(x)R(x). We present an algorithm to compute this decomposition. This allows us to present an algorithm to construct a basis of the cohomology of differential 1-forms with coefficients in Hℚ [x,1/(SD)] for a given H, D being the denominator of dH/H and S ∈ ℚ [x] a square free polynomial. As an application, we generalize a result of Singer on differential equations on the plane: whenever it admits a Liouvillian first integral I but no Darbouxian first integral, our algorithm gives a rational variable change linearising the system.

Let L be a 4th order linear differential operator with coefficients in K(z), with K a computable algebraically closed field. The operator L is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions X satisfies Xt J X=J where J is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if L is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order 4. Moreover, using Klein's Theorem, algebraic solutions are given as pullbacks of standard hypergeometric equations.

Let V ∈ Q(i)(a1,..., an)(q1, q2) be a rationally parametrized planar homogeneous potential of homogeneity degree k ≠ −2, 0, 2. We design an algorithm that computes polynomial necessary conditions on the parameters (a1,..., an) such that the dynamical system associated to the potential V is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9. Another striking application is the first complete proof of the non-integrability of the collinear three body problem.