polarSequence -- Computes the list of multiplcities of a subvareity in the polar varieties of another vareity.

• Usage:

  polarSequence(I)

• Inputs:
  • IX, an ideal, in a polynomial ring $k[x_1,\dots, x_n]$ for a field $k$.
  • IZ, an ideal, in a polynomial ring $k[x_1,\dots, x_n]$ for a field $k$ such that $V(IX)\subset V(IZ)$.
  • polsZ, a list, a list of ideals defining the polar varieties of $Z=V(IZ)$; this is an optional argument to avoid recomputing polar varieties for multiple polar sequence calculations. This is meant to be the output of polarVars(IZ).
• Optional inputs:
  • Algorithm (missing documentation) => ..., default value "saturate",
  • Print (missing documentation) => ..., default value false,
• Outputs:
  • polarMultiplcities, a list of integers giving the (Hilbert-Samuel) multiplicity sequence of $X=V(IX)$ in each of the polar varieties of $Z=V(IZ)$ which contain it.

i1 : R=QQ[x..z]

o1 = R

o1 : PolynomialRing

i2 : IZ=ideal(y^2*z-x^2)

            2     2
o2 = ideal(y z - x )

o2 : Ideal of R

i3 : IX=ideal(x,y,z)

o3 = ideal (x, y, z)

o3 : Ideal of R

i4 : polarSequence(IX, IZ)

o4 = {0, 1, 2}

o4 : List

i5 : polarSequence(ideal(x,y), IZ)

o5 = {0, 0, 2}

o5 : List