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We describe an algebro-geometric construction of polygon-bounded minimalsurfaces in ADS_5, based on consideration of what we call the "boundary ring"of polynomials. The first non-trivial example of the Nambu-Goto (NG) solutionsfor Z_6-symmetric hexagon is considered in some detail. Solutions arerepresented as power series, of which only the first terms are evaluated. TheNG equations leave a number of free parameters (a free function). Boundaryconditions, which fix the free parameters, are imposed on truncated series. Itis still unclear if explicit analytic formulas can be found in this way, buteven approximate solutions, obtained by truncation of power series, can besufficient to investigate the Alday-Maldacena -- BDS/BHT version of thestring/gauge duality.

Boundary Ring: a way to construct approximate NG solutions with polygon boundary conditions: I. Z_n-symmetric configurations