Hyperelliptic action integral

Applying the "exact WKB method" (cf. Delabaere-Dillinger-Pham) to the stati onary one-dimensional Schrodinger equation with polynomial potential, one i s led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties : the projection of its branch-points is in general a dense subset of the plane, and there is a gr oup of symmetries acting on the surface. The distribution of the branch poi nts on the surface is of crucial importance, because it gives the position for the obstacles to Borel-Laplace summation of the WKB-symbols. In "Approc he de la resurgence" by B. Candelpergher, J.-C. Nosmas et F. Pham, p. 103-1 05, an attempt has been made towards giving an explicit construction of the surface with paper, scissors and glue; here we give the correct constructi on and in addition we prove that each surface constructed in this way comes from a polynomial potential. Along the way we are lead to an elementary co njecture in the theory of holomorphic functions.