Boundary ring: A way to construct approximate NG solutions with polygon boundary conditions I. ZnZn-symmetric configurations

We describe an algebro-geometric construction of polygon-bounded minimal surfaces in AdS5AdS5, based on the consideration of what we call the “boundary ring” of polynomials. The first non-trivial example of solutions to the Nambu–Goto (NG) equations for Z6Z6-symmetric hexagon is considered in some detail. Solutions are represented as power series, of which only the first terms are evaluated. The NG equations leave a number of free parameters (a free function). Boundary conditions, which fix the free parameters, are imposed on truncated series. A better use, albeit being exotic to theory of PDE, of the boundary ring is suggested as well. It is still unclear if explicit analytic formulas can be found in this way, but even approximate solutions, obtained by truncation of power series, can be sufficient to investigate the Alday–Maldacena—BDS/BHT version of the string/gauge duality.