irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal

Synopsis

• Usage:

  (ics,p) = irreducibleCharacteristicSeries I

• Inputs:
  • I, an ideal
• Outputs:
  • ics, a list, a list of matrices, representing an irreducible characteristic series for I
  • p, a ring map, an isomorphism from the ring of I to the ring of the characteristic series. The ring retains the names and degrees of the variables, but reorders the variables and uses a default monomial ordering.

Description

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, $lvar(-y w+x^2) = y$ . A triangular set consists of polynomials $f_1,\dots,f_r$ such that $lvar(f_1)< \dots < lvar(f_r)$. In the example, $lvar(-x*y^2+z^3) = x < w = lvar(-w*y+z^2)$ . If $T_1,\dots,T_s$ form an irreducible characteristic series for $I$ , and if $J_i$ is the ideal generated by the largest variables of the elements of $T_i$ , then the algebraic set $V(I)$ defined by $I$ is the union of the sets $V(T_i) \setminus V(I_i)$, for $i=1,\dots,s$. 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

See also

• minimalPrimes -- minimal primes of an ideal