fSplittingHeight -- Finds the quasi-F-split height ht(S/I) of the quotient of the polynomial ring S=(ZZ/p)[x1,...,xn] by an ideal I generated by a regular sequence (f1,...,fm).

Usage:

fSplittingHeight I
Inputs:
- I, an ideal,
Optional inputs:
- MaxHeight => ..., default value 102
Outputs:
- r, a number,

Description

This gives the quasi-F-Splitting height r of S/I

i1 : S = ZZ/3[x,y,z, w]

o1 = S

o1 : PolynomialRing

i2 :         I = ideal(x^4 + y^4 + z^4 + w^4 + x^2*z^2 + x*y*z^2)

            4    4    2 2        2    4    4
o2 = ideal(x  + y  + x z  + x*y*z  + z  + w )

o2 : Ideal of S

This should give 4.

One can change the max F-splitting height that the algorithm tests

i3 : fSplittingHeight I
trying i = 2
trying i = 3
trying i = 4

o3 = 4

i4 : S = ZZ/3[x,y,z, w]

o4 = S

o4 : PolynomialRing

This should not return anything.

i5 :     I = ideal(x^4 + y^4 + z^4 + w^4 + x^2*z^2 + x*y*z^2)

            4    4    2 2        2    4    4
o5 = ideal(x  + y  + x z  + x*y*z  + z  + w )

o5 : Ideal of S

i6 : fSplittingHeight (I,MaxHeight=>2)
trying i = 2

i7 : S = ZZ/3[x,y,z,w]

o7 = S

o7 : PolynomialRing

This should give infinity.

i8 : I = ideal(x^4 + y^4 + z^4 + w^4)

            4    4    4    4
o8 = ideal(x  + y  + z  + w )

o8 : Ideal of S

This should give 1.

i9 : fSplittingHeight I

o9 = infinity

o9 : InfiniteNumber

i10 : S = ZZ/3[x,y,z]

o10 = S

o10 : PolynomialRing

i11 :     I = ideal(x,y)

o11 = ideal (x, y)

o11 : Ideal of S

i12 : S = ZZ/5[x,y,z]

o12 = S

o12 : PolynomialRing

Here, we would get an error since the ideal is not generated by a regular sequence

i13 : I = ideal(y*x,z*x)

o13 = ideal (x*y, x*z)

o13 : Ideal of S

i14 : stopIfError = false

o14 = false

i15 : fSplittingHeight I
stdio:15:16:(3):[1]: error: expected ideal to be a complete intersection

i16 : S = GF(4)[x,y,z]

o16 = S

o16 : PolynomialRing

Here, we would get an error since S is not a polynomial ring over ZZ/p.

Ways to use fSplittingHeight:

fSplittingHeight(Ideal) (missing documentation)

For the programmer

The object fSplittingHeight is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/WittVectors/Documentation.m2:410:0.