Here are some notes I took when I tried to allow expressions ove finite fields.

when making ZMOD an OrderedSet in the obvious way, I can have EXPR ZMOD 5 and EXPR PF 5. However, not all things work that should:

  a:EXPR PF 5 := 2
  a < 3$EXPR PF 5 

gives false, as does 3$EXPR PF 5 < a

The reason for this is that in polycat.spad/POLYCAT, the following definition is used. It should be OrderedStructure, maybe:

    if R has OrderedSet then
      p:% < q:% ==
        (dp:= degree p) < (dq := degree q) => (leadingCoefficient q) > 0
        dq < dp => (leadingCoefficient p) < 0
--        leadingCoefficient(p - q) < 0
-- the last test works only if < is compatible with -
-- the following works, but is slower, of course
        leadingCoefficient(p) < leadingCoefficient(q)

Furthermore, saying:

  a:EXPR PF 3 := 2
  eval(a^x,x=5)

gives 2^2 unevaluated.

Here we have two problems at once:

• it should be 2^5=32=2 rather than 2^2=4=1 (that is, it treats x as an element of EXPR PF 3 rather than an integer.

  Question: is a^(b::EXPR PF 3) meaningful? Rather not, since the power laws do not hold: 2=2*1=2^1*2^2=2^3=2^0=1 (3) What is required so that a^b is well defined?

• it should evaluate the expression.

  This happens in COMBF, since... Hmmm, I didn't take notes here anymore. I probably didn't find out.