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FThresholds :: FThresholds

FThresholds -- a package for computing numerical invariants in prime characteristic

Description

The Frobenius endomorphism on a ring of prime characteristic p, which sends a ring element to its p-th power, is a fundamental tool in prime characteristic commutative algebra. Kunz has shown that regularity is characterized by the behavior of this map, and since then, many other properties of Frobenius have been used to measure how far a ring is from being regular.

Numerical invariants of rings, and of their elements and ideals, play an important role in this endeavor. Of particular focus are the F-pure threshold, and more generally, F-thresholds.

However, the computation of these numerical invariants can be quite subtle, and at the same time computationally complex. In partnership with the package TestIdeals, which provides some important functionality for researchers in positive characteristic commutative algebra, the package FThresholds implements known algorithms, as well as new methods, to estimate and compute numerical invariants in prime characteristic.

See also

Authors

Version

This documentation describes version 1.0 of FThresholds.

Source code

The source code from which this documentation is derived is in the file FThresholds.m2. The auxiliary files accompanying it are in the directory FThresholds/.

Exports

  • Functions and commands
    • compareFPT -- checks whether a given number is less than, greater than, or equal to the F-pure threshold
    • criticalExponentApproximation -- gives a list of approximates of a critical exponent
    • fpt -- attempts to compute the F-pure threshold of a polynomial at the origin
    • fptApproximation -- gives a list of terms in the sequence whose limit defines the F-pure threshold
    • ftApproximation -- gives a list of terms in the sequence whose limit defines an F-threshold
    • isFJumpingExponent -- Checks whether a given number is an F-jumping number
    • isFPT -- Checks whether a given number is the F-pure threshold
    • mu -- computes the largest Frobenius power of an ideal not contained in a specified Frobenius power
    • muList -- computes a list of mu-values associated to a given F-threshold or F-pure threshold
    • nu -- computes the largest power of an ideal not contained in a specified Frobenius power
    • nuList -- computes a list of nu-values associated to a given F-threshold or F-pure threshold
  • Symbols
    • Attempts (missing documentation) -- generates an inhomogeneous system of parameters
    • BinaryRecursive -- an option value specifying a binary recursive search method
    • ComputePreviousNus -- an option to compute nu-values recursively
    • ContainmentTest -- an option to specify the containment test used
    • FRegularityCheck -- an option to use an F-regularity check to find an F-pure threshold
    • FrobeniusPower -- an option value to consider containment of Frobenius powers of ideals
    • FrobeniusRoot -- an option value to consider containment of Frobenius roots of ideals
    • MaxChecks -- specifies the number of "guess and check" attempts to make in an F-pure threshold computation
    • Search -- an option to specify the search method
    • StandardPower -- an option value to consider containment of standard power of an ideal in Frobenius power of another ideal
    • UseColonIdeals -- an option to use colon ideals to compute nus in an iterative way
    • UseFSignature -- whether to use the F-signature function in the search for an F-pure threshold
    • UseSpecialAlgorithms -- an option to check whether the input is a diagonal polynomial, binomial, or binary form