mu -- computes the largest Frobenius power of an ideal not contained in a specified Frobenius power

Synopsis

• Usage:
mu(e,I,J)
mu(e,I)
mu(e,f,J)
mu(e,f)
ComputePreviousNus => Boolean
Search => Symbol
UseColonIdeals => Boolean
• Inputs:
• Optional inputs:
• ComputePreviousNus => , default value true, specifies whether to compute nu(d,I,J) for d = 0, …, e-1 to aid in the computation of mu(e,I,J)
• Search => , default value Binary, specifies the strategy in which to search for the largest integer n such that the n-th generalized Frobenius power of I is not contained in some specified Frobenius power of J.
• UseColonIdeals => , default value false, specifies whether to use colon ideals in a recursive manner when computing mu(e,I,J)
• Outputs:
• an integer, the e-th value μ associated to the F-threshold or F-pure threshold

Description

Given an ideal I in a polynomial ring k[x1, ..., xn], mu(e, I, J) or mu(e, f, J) outputs the maximal integer N such that the N-th generalized Frobenius power of I, or fN, is not contained in the pe-th Frobenius power of J.

 `i1 : R = ZZ/3[x,y];` ```i2 : I = ideal(x^2, x+y); o2 : Ideal of R``` ```i3 : J = ideal(x, y^2); o3 : Ideal of R``` ```i4 : mu(2,I,J) o4 = 17``` ```i5 : mu(3,I) o5 = 26``` ```i6 : mu(3,x^3+y^3,J) o6 = 17```