1.Build the Sylvester matrix corresponding to the polynomials in the sub-resultant lecture, i.e:
Run Bareiss’s algorithm on this to compute the determinant. Note that you have to deal with the case of zero elements on the diagonal.
2.Emulate the Buchberger algorithm on cyclic-3, i.e.:
By this I mean that each s-polynomial should be computed and reduced under human control, i.e. the most sophisticated MAPLE commands you can use are of the form s:= 28x*f1-3*y*f2 or S:= s-7*z*f3. how many S-polynomials do you compute? How many of them reduce to zero?
3.Compute, via the Faugère-Gianni-Lazard-Mora(FGLM) algorithm, a purely lexicographical Gröbner base for the cyclic-5 problem: i.e:
Hence deduce the number of solutions and a description of them.