isFPT -- checks whether a given rational number is the F-pure threshold

Synopsis

• Usage:

  isFPT(t, f)

• Inputs:
  • t, a number, a rational number candidate for the $F$-pure threshold of f
  • f, a ring element, in a $\mathbb{Q}$-Gorenstein ring of positive characteristic $p$, whose index is not divisible by $p$
• Optional inputs:
  • AssumeDomain => a Boolean value, default value true, indicates whether the ambient ring of f is an integral domain
  • FrobeniusRootStrategy => a symbol, default value Substitution, passed to computations in the TestIdeals package
  • AtOrigin => a Boolean value, default value false, tells the function whether to consider only the behavior at the origin
  • MaxCartierIndex => an integer, default value 10, sets the maximum $\mathbb{Q}$-Gorenstein index to search for
  • QGorensteinIndex => an integer, default value 0, specifies the $\mathbb{Q}$-Gorenstein index of the ring
  • Verbose => a Boolean value, default value false, whether the output is to be verbose
• Outputs:
  • a Boolean value, reporting whether t is the $F$-pure threshold of f

Description

i1 : R = ZZ/11[x,y];

i2 : f = x^3 + y^2;

i3 : isFPT(9/11, f)

o3 = true

i4 : isFPT(9/12, f)

o4 = false

i5 : T = ZZ/7[a,b];

i6 : S = ZZ/7[x,y,z,w];

i7 : f = map(T, S, {a^3, a^2*b, a*b^2, b^3});

o7 : RingMap T <-- S

i8 : R = S/ker f;

i9 : isFPT(1/3, x)

o9 = true

i10 : isFPT(1/3 + 1/10000, x)

o10 = false

i11 : isFPT(1/3 - 1/10000, x)

o11 = false

i12 : R = ZZ/11[x,y,z]/(x^2 - y*(z - 1));

i13 : isFPT(1/2, z - 1)

o13 = true

i14 : isFPT(1/2, z - 1, AtOrigin => true)

o14 = false

See also

• compareFPT -- determine whether a number is less than, greater than, or equal to the F-pure threshold
• fpt -- attempt to compute the F-pure threshold of a polynomial at the origin or globally
• isFJumpingExponent -- whether a given number is an F-jumping exponent