As we see in the example below, an irreducible characteristic series for I consists of a collection of triangular sets. Here, given a polynomial f, write lvar(f) for the largest variable appearing in f (with respect to the lexicographic order). In the example, lvar(-y w+x2) = y . A triangular set consists of polynomials f1,…,fr such that lvar(f1)< …< lvar(fr). In the example, lvar(-x*y2+z3) = x < w = lvar(-w*y+z2) . If T1,…,Ts form an irreducible characteristic series for I , and if Ji is the ideal generated by the largest variables of the elements of Ti , then the algebraic set V(I) defined by I is the union of the sets V(Ti) \V(Ii), for i=1,…,s. The minimal associated primes of I can thus be recovered from the irreducible characteristic series by saturation and by throwing away superfluous primes. This is done by minimalPrimes, which uses this routine.
i1 : R = QQ[w,x,y,z]; |
i2 : (L,p) = irreducibleCharacteristicSeries ideal(x^2-y*w,x^3-z*w^2) o2 = ({| -zw2+x3 -yw+x2 |, | x w |}, map(R,QQ[y, z, w, x],{y, z, w, x})) o2 : Sequence |
i3 : apply(L, m -> p m) o3 = {| x3-w2z x2-wy |, | x w |} o3 : List |
i4 : p^-1 o4 = map(QQ[y, z, w, x],R,{w, x, y, z}) o4 : RingMap QQ[y, z, w, x] <--- R |