Description
FrobeniusThresholds is a package for computing
Fpure thresholds,
Fjumping exponents, and related numerical measures of singularity in positive characteristic.
The Frobenius endomorphism on a ring of prime characteristic
p > 0, which sends a ring element to its
pth power, is a fundamental tool in positive characteristic commutative algebra. Kunz showed that regularity is characterized by the behavior of this map, and since then many other properties of the Frobenius map have been used to measure the 'severity' of singularities in commutative algebra. The
Fpure threshold is a prominent object in this realm.
This package relies heavily on the
TestIdeals package. Many special cases (binomials, diagonal polynomials, etc.) are
seamlessly handled by using special algorithms found in
Hernández,
Hernández, and
HernándezTeixeira.
Notable functions:

fpt computes (or estimates) the Fpure threshold.

isFJumpingExponent checks whether a given number is an Fjumping exponent.

compareFPT determines whether the given number is greater than, less than, or equal to the Fpure threshold.

nu is a function whose normalized value provides a canonical estimate for the Fpure threshold, Fthresholds, and more.
The following example demonstrates some of the functionality of this package.
i1 : p = 131;

i2 : R = ZZ/p[x,y];

i3 : f = x^13  y^5;

i4 : c = fpt(f)
18
o4 = 
65
o4 : QQ

i5 : compareFPT(c  1/p^2, f)
o5 = 1

i6 : compareFPT(c, f)
o6 = 0

i7 : compareFPT(c + 1/p, f)
o7 = 1

i8 : isFJumpingExponent(36/65, f)
o8 = true

Acknowledgements:The authors would like to thank David Eisenbud, Daniel Grayson, Anurag Singh, Greg Smith, and Mike Stillman for useful conversations and comments on the development of this package.
ContributorsWe sincerely thank the following people, who contributed code to this package.