Depth off-by-one when ring includes "a" as variable

[anna-brosowsky]

When using the depth function from the Depth package, the depth of a ring seems to be one larger than it is supposed to be whenever the ring is a quotient ring of a polynomial ring that includes "a" as a variable. In particular, it is sometimes giving that depth is larger than dimension! Some small examples:

i1 : needsPackage("Depth");
i2 : depth(GF(2)[x]/ideal(x^2))
o2 = 0
i3 : depth(GF(2)[a]/ideal(a^2))
o3 = 1
i4 : depth(GF(2)[b]/ideal(b^2))
o4 = 0

A larger example:

i5 : depth(GF(5)[x,y,z]/ideal(x^3+y^3+z^3))
o5 = 2
i6 : depth(GF(5)[a,b,c]/ideal(a^3+b^3+c^3))
o6 = 3

But notice a polynomial ring itself using "a" works fine:

i7 : depth(GF(5)[a,b,c])
o7 = 3

This issue of course carries down to other functions that use depth, such as isCM.

[d-torrance]

I think the problem isn't with the Depth package, but with Galois fields. It works just fine with other coordinate fields:

i2 : depth(QQ[a]/ideal(a^2))

o2 = 0

i3 : depth(ZZ/2[a]/ideal(a^2))

o3 = 0

The following code is likely to blame: https://github.com/Macaulay2/M2/blob/ec9e9ac60ed4a8e791448942202f077a88a87e15/M2/Macaulay2/m2/galois.m2#L99

[d-torrance]

After playing around with this, I think this is actually the intended behavior, but the documentation for GF should probably be improved to explain what's going on.

If GF is called without the Variable option, then a is set globally as a generator of the field. So we shouldn't be using it as a variable for a polynomial ring over the field.

i1 : GF 2

o1 = GF 2

o1 : GaloisField

i2 : a

o2 = 1

o2 : GF 2

[d-torrance]

@ab2422 - Do you think the fix in #3259 (adding documentation as to why the variable a is special when using Galois fields) is sufficient to close this issue?

[d-torrance]

Related: #897

[anna-brosowsky]

@d-torrance Sure, the documentation fix is helpful & seems good enough to close. Thanks for looking at it!