Algebraic independence of the periods of Abelian integrals (Q1387389)

The author considers an irreducible algebraic curve of genus \(g\leq 1\) over the field of algebraic numbers \(\mathbb{A}\). On the Riemann surface of the curve there exist \(2g\) Abelian differentials of the second kind \(\phi_1,\dots, \phi_{2g}\) over \(\mathbb{A}\) and \(2g\) cycles \(\gamma_1,\dots, \gamma_{2g}\) such that the matrix \(\Omega= (\omega_{i,j})_{i,j=1,\dots, 2g}\) of their periods \[ \omega_{i,j}= \int_{\gamma_j} \phi_i, \quad i,j= 1,\dots, 2g \] is nondegenerate. The following theorem is proven in the paper: Every \(g+1\) different rows of the matrix \(\Omega\) contain at least two algebraically independent elements. The theorem is applied to the hyperelliptic curve \(y^2= 1-x^n\) to prove algebraic independence of certain values of the Euler \(\Gamma\)-function.