Symbolic integration of hyperexponential 1-forms
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Let H be a hyperexponential function in n variables x=(x_1,…,x_n) with coefficients in a field 𝕂, [𝕂:ℚ] <∞, and ω a rational differential 1-form. Assume that Hω 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 H𝕂[x,1/(SD)] for a given H, D being the denominator of dH/H and S∈𝕂[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.
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