Edit detail for finite field expressions revision 1 of 1

changed:
-
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.

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

    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)

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