RFC: Maximal ideal in FriCAS

[Grégory Vanuxem]

Hello,

To continue, here is an implementation in PolynomialIdeal (file attached), to check if an ideal is maximal. I used Gemini first, checked its code manually (shape lemma for example that I didn't know), fixed 3 groebner basis computations to 1 computation for example, some improvements/issues remain I guess, and finally asked Claude to check the code. The algorithm is not totally deterministic for multi-variables (5 random attempts, Tschirnhaus) and uses the shape lemma that you should know. The rest is academic I think.

I assume it could be improved, and it might even contain some bugs. If you have any comments, improvements or fixes, I would be happy to read them.

Greg

     maximal?(I : Ideal, lvar : List VarSet) : Boolean ==

       element?(1, I) => false

       empty? lvar => empty? (groebner I).idl

       not zeroDim?(I, lvar) => false

       n : NNI := #lvar

       J := groebner I

       Jd := J.idl

       n = 1 =>

         empty? Jd => false

         g := Jd.first

         x := lvar.first

         deg := degree(univariate(g, x))

         deg = 0 => false

         deg = 1 => true

         if F has PolynomialFactorizationExplicit then -- TODO: implement for Nemo

           fl := factorList(factor(g)$DPoly)

           #fl = 1 and fl(1).exponent = 1

         else

           false

       x := lvar.last

       lv1 := remove(x, lvar)

       for attempt in 1..5 repeat

         if #Jd = n then

           ok : Boolean := true

           i : NNI := 1

           for f in Jd while ok repeat

             if i < n then

               if not isMonic?(f, lvar(i)) or degree(univariate(f, lvar(i))) ~= 1 then

                 ok := false

             i := i + 1

           if ok then

             g := Jd(n)

             deg := degree(univariate(g, x))

             deg = 0 => return false

             deg = 1 => return true

             if F has PolynomialFactorizationExplicit then

               fl := factorList(factor(g)$DPoly)

               return (#fl = 1 and fl(1).exponent = 1)

             else

               return false

         ranvals : List Z := [random(23)$Z for vv in lv1] -- use #lv1 ?

         val : DPoly := _+/[rv * (vv::DPoly) for vv in lv1 for rv in ranvals] + (x::DPoly)

         J := groebnerIdeal groebner [(univariate(p, x)).val for p in J.idl]

         Jd := J.idl

       false

     maximal?(I : Ideal) : Boolean == maximal?(I, "setUnion"/[variables g for g in I.idl])