isFRationalThreshold -- determines if a given number is the F-rational threshold, or, equivalently in the case of a quasi-Gorenstein ring, the F-pure threshold

• Usage:

  b = isFRationalThreshold(t, I)

  b = isFPT(t,I)

• Inputs:
  • t, a rational number, the number which might equal the threshold
  • I, an ideal, the ideal whose threshold you want to compute
• Optional inputs:
  • AtOrigin => ..., default value false
  • FrobeniusRootStrategy => ..., default value Substitution
  • Verbose => ..., default value false
• Outputs:
  • b, a Boolean value, is it the threshold or not

i1 : R = ZZ/3[x,y,z];

i2 : I = ideal(x^2,y^3,z^5); --fpt should be 1/2 + 1/3 + 1/5 = 31/30

o2 : Ideal of R

i3 : isFPT(31/30, I)

o3 = true

i4 : isFRationalThreshold(31/30, I)

o4 = true

i5 : isFRationalThreshold(1, I)

o5 = false

i6 : isFPT(32/30, I)

o6 = false

i7 : R = ZZ/3[u,v]

o7 = R

o7 : PolynomialRing

i8 : I = ideal( (u-1)^2, v^3) --fpt should be 1/2+1/3 = 5/6, but it's not the threshold at the origin

             2           3
o8 = ideal (u  + u + 1, v )

o8 : Ideal of R

i9 : isFPT(5/6, I)

o9 = true

i10 : isFRationalThreshold(5/6, I, AtOrigin=>true)

o10 = false