--Example 1 restart R=QQ[x..z] f=det(matrix{{1,x,y},{x,1,z},{y,z,1}}) --the singular points are J=ideal(jacobian(ideal f))+ideal(f) decompose J --the plot is the following projective transformation gr=-1*sub(f,{x=>(3)*x/(1-x-y-z),y=>(3)*y/(1-x-y-z),z=>(3)*z/(1-x-y-z)}) g=numerator gr --Example 2 restart R=QQ[x] I=ideal(x^3+6*x^2+12*x+8) J=ideal((x-1)*(x+2)) factor I_0 factor J_0 --Intersection K=intersect(I,J) factor K_0 --Product K=I*J factor K_0 --Sum g=gcd(I_0,J_0) G=ideal groebnerBasis(I+J) (I+J)==G --Quotient/Colon K=I:J factor K_0 factor I_0 factor J_0 K=J:I factor K_0 --Example 3 restart R=QQ[x,y] I=ideal(x^2+10*x+34,3*y-2*x-13) S=R/I f=sub(1/13*(x+5*y),S) f^2 isPrime I radical(I)==I isPrimary I --Example 4 restart R=QQ[x..z] f=det(matrix{{1,x,y},{x,1,z},{y,z,1}}) --the partial dirivities are J=ideal(jacobian(ideal f)) --note f is not in the ideal J since f has constant --term =1 where as everything in J has constant term -- zero isSubset(ideal f, J) decompose J -- the first point is NOT on the surface -- V(f), but the other 4 points are decompose(J+ideal(f)) --geometrically adding ideals coresponds --to intersecting zero sets