Many problems of confounding and identifiability for polynomial models in an experimental design can be solved using methods of algebraic geometry. The theory of $Grddot{o}bner$ basis is used to characterize the design. In addition, a fractional factorial design can be uniquely represented by a polynomial indicator function. $Grddot{o}bner$ bases and indicator functions are powerful computational tools to deal with ideals of fractions based on each different theoretical aspects. The problem posed here is to give how to move from one representation to the other. For a given fractional factorial design, the indicator function can be computed from the generating equations in the $Grddot{o}bner$ basis. The theory is tested using some fractional factorial designs aided by a modern computational algebra package CoCoA.