  We describe an algebro-geometric construction of polygon-bounded minimal
surfaces 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) solutions
for Z_6-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. 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.
