A SAGBI BASIS FOR SOME SUBALGEBRAS OFrnPOLYNOMIAL RINGSrnDawit Solomon TadessernAddis Ababa University, 2018rnThe term SAGBI" is an acronym for Subalgebra Analogue to Gr obner Basesrnfor Ideals". There exist _nitely generated subalgebras of |[x1; x2; :::; xn] whichrnhave no _nite SAGBI basis with respect to any monomial order. There are alsornsubalgebras which may or may not have _nite SAGBI basis depending on thernmonomial order de_ned on |[x1; x2; :::; xn]. We know that, there are uncountablyrnmany monomial orders de_ned on |[x1; x2; :::; xn] for n _ 2. It is still an importantrnopen problem to classify subalgebras that have a _nite SAGBI basis. In additionrnto this, two or more generators of a subalgebra may not form a SAGBI basis.rnTorstensson et al. provide su_cient and necessary conditions for two generatorsrnform a SAGBI basis, in the case of univariate polynomial ring. An other importantrnopen problem is to provide a su_cient andnor necessary condition when three orrnmore generators form a SAGBI basis in the univariate polynomial ring as wellrnas two or more generators form a SAGBI basis in a multivariate polynomial ring.rnThis thesis provides su_cient conditions when two generators of a subalgebra formrna SAGBI basis in a multivariate polynomial ring. We also investigate su_cientrnand necessary condition when generators with consecutive degrees form a SAGBIrnbasis in the univariate polynomial ring. Finally we conjecture that subalgebrasrngenerated by two polynomials have a _nite SAGBI basis with respect to anyrnmonomial order. We prove our conjecture in certain cases.