Solving polynomial system is a basic problem in the field of computational sciences and engineering . Grobner bases is a powerful tool in solving non-linear polynomial systems over different kind of fields . However , the complexity of Grobner bases computation lies in the worst complexity The characteristic set method of Wu has freed Ritt's decomposition from polynomial factorization , opening the door to a variety of discoveries in polynomial and differential algebra . During the past 25 years the work of Wu has been extended to allow for more powerful decompositon algorithms and applied to different types of polynomial systems . Today , triangular decompositon algorithms are available in several software package system solvers , such as MAPLE'S solve command. Incremental algorithms for triangular decomposition rely on a procedure for computing , the intersection of a hypersurface and the quasi-component of regular chain . Thus , the input of this operation can be regarded as well-behaved geometrical objects . However , known algorithms , namely the one of Lazard and the one of Moreno Maza are quite involved and difficult to analyze and optimize . Let us illustrate the main algorithm (with an incremental strategy) of this thesis the following example . Consider a polynomial system F={p 1 ,p 2 ,p 3 } in Q[x,y,z] where x y z and p 1 =x 2 +y 2 -z 2 -1 , p 2 =x 2 -y 2 -z 2 -1 , p 3 =z 3 +xy-1 . A triangular decomposition of F can be computed incrementally as follows . We first compute a triangular decompusition of p 1 by calling Triangularize(p 1 ) . The output simply consist of one regular chain T 1 = {p 1 } Next we compute a triangular decomposition of {p 1 ,p 2 }, which is achived by calling Intersect(p 2 ,T 1 ) whose output consists of one regular chain T 2 , where T 2 ={2z 3 -3,2y 2 +2x 2 -5}. Finally we compute a triangular decomposition of F by calling $Intersect(p 3 ,T 2 ) , which consists of a regular chain T 3 , where T 3 ={3z+2xy-2,16xy+8x 4 -20x 2 +19,64x 8 -320x 6 +960x 4 -1400x 2 +361}. The zero set of T 3 is exactly the zero set of F, but since it has a triangular form, we can solve T 3 simpler than F .