)abbrev domain GDMP GeneralDistributedMultivariatePolynomial
++ Author: Barry Trager
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd, leadingCoefficient
++ Related Constructors: DistributedMultivariatePolynomial,
++ HomogeneousDistributedMultivariatePolynomial
++ Also See: Polynomial
++ AMS Classifications:
++ Keywords: polynomial, multivariate, distributed
++ References:
++ Description:
++   This type supports distributed multivariate polynomials
++ whose variables are from a user specified list of symbols.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
++ The term ordering is specified by its third parameter.
++ Suggested types which define term orderings include: \spadtype{DirectProduct},
++ \spadtype{HomogeneousDirectProduct}, \spadtype{SplitHomogeneousDirectProduct}
++ and finally \spadtype{OrderedDirectProduct} which accepts an arbitrary user
++ function to define a term ordering.

GeneralDistributedMultivariatePolynomial(vl, R, E) : public == private where
  vl : List Symbol
  R : Ring
  E : DirectProductCategory(#vl, NonNegativeInteger)
  OV  ==> OrderedVariableList(vl)
  SUP ==> SparseUnivariatePolynomial
  NNI ==> NonNegativeInteger

  public == PolynomialCategory(R, E, OV) with
      reorder : (%, List Integer) -> %
        ++ reorder(p, perm) applies the permutation perm to the variables
        ++ in a polynomial and returns the new correctly ordered polynomial

  private == PolynomialRing(R, E) add
    --representations
      Term := Record(k : E, c : R)
      Rep := List Term
      n := #vl
      Vec ==> Vector(NonNegativeInteger)
      zero?(p : %) : Boolean == empty?(p::Rep)

      totalDegree p ==
         zero? p => 0
         "max"/[reduce("+",(t.k)::(Vector NNI), 0) for t in p]

      monomial(p : %, v : OV, e : NonNegativeInteger) : % ==
         locv := lookup v
         p*monomial(1,
            directProduct [if z = locv then e else 0 for z in 1..n]$Vec)

      coerce(v : OV) : % == monomial(1, v, 1)

      listCoef(p : %) : List R ==
        rec : Term
        [rec.c for rec in (p::Rep)]

      mainVariable(p : %) ==
         zero?(p) => "failed"
         for v in vl repeat
           vv := variable(v)::OV
           if degree(p, vv)>0 then return vv
         "failed"

      ground?(p) == mainVariable(p) case "failed"

      retract(p : %) : R ==
          not ground? p => error "not a constant"
          leadingCoefficient p

      retractIfCan(p : %) : Union(R,"failed") ==
        ground?(p) => leadingCoefficient p
        "failed"

      degree(p : %, v : OV) ==
         -- degree(univariate(p, v))
         zero?(p) => 0
         res : NonNegativeInteger := 0
         locv := lookup v
         while not empty? p repeat
             t := first p
             j := t.k.locv
             if j > res then res := j
             p := rest p
         res

      minimumDegree(p : %, v : OV) == minimumDegree(univariate(p, v))
      differentiate(p : %, v : OV) ==
            multivariate(differentiate(univariate(p, v)), v)

      degree(p : %, lv : List OV) == [degree(p, v) for v in lv]
      minimumDegree(p : %, lv : List OV) == [minimumDegree(p, v) for v in lv]

      numberOfMonomials(p : %) ==
        l : Rep := p :: Rep
        empty?(l) => 1
        #l

      monomial?(p : %) : Boolean ==
        l : Rep := p :: Rep
        empty?(l) or empty?(rest(l))

      if R has OrderedRing then
        maxNorm(p : %) : R ==
          l : List R := []
          r, m : R
          m := 0
          for r in listCoef(p) repeat
            if r > m then m := r
            else if (-r) > m then m := -r
          m

      --trailingCoef(p : %) ==
      --  l : Rep := p : Rep
      --  empty?(l) => 0
      --  r : Term := last l
      --  r.c

      --leadingPrimitiveMonomial(p : %) ==
      --  ground?(p) => 1$%
      --  r : Term := first(p: Rep)
      --  r := [r.k, 1$R]$Term     -- new cell
      -- list(r)$Rep :: %

    -- The following 2 defs are inherited from PolynomialRing

      --leadingMonomial(p : %) ==
      --  ground?(p) => p
      --  r : Term := first(p: Rep)
      --  r := [r.k, r.c]$Term     -- new cell
      --  list(r)$Rep :: %

      --reductum(p : %): % ==
      --  ground? p => 0$%
      --  (rest(p: Rep)): %

      if R has Field then
        (p : %) / (r : R) == inv(r) * p

      variables(p : %) ==
         maxdeg : Vector(NonNegativeInteger) := new(n, 0)
         while not zero?(p) repeat
            tdeg := degree p
            p := reductum p
            for i in 1..n repeat
              maxdeg.i := max(maxdeg.i, tdeg.i)
         [index(i::PositiveInteger) for i in 1..n | maxdeg.i ~= 0]

      reorder(p : %, perm : List Integer) : % ==
         #perm ~= n => error "must be a complete permutation of all vars"
         q := [[directProduct [term.k.j for j in perm]$Vec, term.c]$Term
                         for term in p]
         sort((z1, z2) +-> z1.k > z2.k, q)

      --coerce(dp: DistributedMultivariatePolynomial(vl, R)): % ==
      --   q := dp: List(Term)
      --   sort(#1.k > #2.k, q): %

      univariate(p : %, v : OV) : SUP(%) ==
         zero?(p) => 0
         exp := degree p
         locv := lookup v
         deg : NonNegativeInteger := 0
         nexp := directProduct [if i = locv then (deg := exp.i;0) else exp.i
                                        for i in 1..n]$Vec
         monomial(monomial(leadingCoefficient p, nexp), deg)+
                      univariate(reductum p, v)

      eval(p : %, v : OV, val : %) : % == univariate(p, v)(val)

      eval(p : %, v : OV, val : R) : % == eval(p, v, val::%)$%

      eval(p : %, lv : List OV, lval : List R) : % ==
         lv = [] => p
         eval(eval(p, first lv, (first lval)::%)$%, rest lv, rest lval)$%

      -- assume Lvar are sorted correctly
      evalSortedVarlist(p : %, Lvar : List OV, Lpval : List %) : % ==
        v := mainVariable p
        v case "failed" => p
        pv := v:: OV
        Lvar=[] or Lpval=[] => p
        mvar := Lvar.first
        mvar > pv => evalSortedVarlist(p, Lvar.rest, Lpval.rest)
        pval := Lpval.first
        pts : SUP(%) := map(x+->evalSortedVarlist(x, Lvar, Lpval), univariate(p, pv))
        mvar = pv => pts(pval)
        multivariate(pts, pv)

      eval(p : %, Lvar : List OV, Lpval : List %) ==
        nlvar : List OV := sort((x, y) +-> x > y, Lvar)
        nlpval :=
           Lvar = nlvar => Lpval
           nlpval := [Lpval.position(mvar, Lvar) for mvar in nlvar]
        evalSortedVarlist(p, nlvar, nlpval)

      multivariate(p1 : SUP(%), v : OV) : % ==
        0 = p1 => 0
        degree p1 = 0 => leadingCoefficient p1
        leadingCoefficient(p1)*(v::%)^degree(p1) +
                  multivariate(reductum p1, v)

      univariate(p : %) : SUP(R) ==
        (v := mainVariable p) case "failed" =>
                      monomial(leadingCoefficient p, 0)
        q := univariate(p, v:: OV)
        ans : SUP(R) := 0
        while q ~= 0 repeat
          ans := ans + monomial(ground leadingCoefficient q, degree q)
          q := reductum q
        ans

      multivariate(p : SUP(R), v : OV) : % ==
        0 = p => 0
        (leadingCoefficient p)*monomial(1, v, degree p) +
                       multivariate(reductum p, v)

      if R has GcdDomain then
        content(p : %) : R ==
          zero?(p) => 0
          "gcd"/[t.c for t in p]



        if R has EuclideanDomain and not(R has FloatingPointSystem)  then
          gcd(p : %, q : %) : % ==
            gcd(p, q)$PolynomialGcdPackage(E, OV, R, %)

        else gcd(p : %, q : %) : % ==
            r : R
            (pv := mainVariable(p)) case "failed" =>
              (r := leadingCoefficient p) = 0$R => q
              gcd(r, content q)::%
            (qv := mainVariable(q)) case "failed" =>
              (r := leadingCoefficient q) = 0$R => p
              gcd(r, content p)::%
            pv<qv => gcd(p, content univariate(q, qv))
            qv<pv => gcd(q, content univariate(p, pv))
            multivariate(gcd(univariate(p, pv), univariate(q, qv)), pv)

      coerce(p : %) : OutputForm ==
        zero?(p) => (0$R) :: OutputForm
        l, lt : List OutputForm
        lt := []
        vl1 := [v::OutputForm for v in vl]
        for t in reverse p repeat
          l := []
          for i in 1..#vl1 repeat
            t.k.i = 0 => "next"
            t.k.i = 1 => l := cons(vl1.i, l)
            l := cons(vl1.i ^ t.k.i ::OutputForm, l)
          l := reverse l
          if (t.c ~= 1) or empty?(l) then l := cons(t.c :: OutputForm, l)
          1 = #l => lt := cons(first l, lt)
          lt := cons(reduce("*",l),lt)
        1 = #lt => first lt
        reduce("+",lt)

)abbrev domain DMP DistributedMultivariatePolynomial
++ Author: Barry Trager
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd, leadingCoefficient
++ Related Constructors: GeneralDistributedMultivariatePolynomial,
++ HomogeneousDistributedMultivariatePolynomial
++ Also See: Polynomial
++ AMS Classifications:
++ Keywords: polynomial, multivariate, distributed
++ References:
++ Description:
++   This type supports distributed multivariate polynomials
++ whose variables are from a user specified list of symbols.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
++ The term ordering is lexicographic specified by the variable
++ list parameter with the most significant variable first in the list.
DistributedMultivariatePolynomial(vl, R) : public == private where
  vl : List Symbol
  R  : Ring
  E   ==> DirectProduct(#vl, NonNegativeInteger)
  OV  ==> OrderedVariableList(vl)
  public == PolynomialCategory(R, E, OV) with
      reorder : (%, List Integer) -> %
        ++ reorder(p, perm) applies the permutation perm to the variables
        ++ in a polynomial and returns the new correctly ordered polynomial

  private ==
    GeneralDistributedMultivariatePolynomial(vl, R, E)

)abbrev domain HDMP HomogeneousDistributedMultivariatePolynomial
++ Author: Barry Trager
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd, leadingCoefficient
++ Related Constructors: DistributedMultivariatePolynomial,
++ GeneralDistributedMultivariatePolynomial
++ Also See: Polynomial
++ AMS Classifications:
++ Keywords: polynomial, multivariate, distributed
++ References:
++ Description:
++   This type supports distributed multivariate polynomials
++ whose variables are from a user specified list of symbols.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
++ The term ordering is total degree ordering refined by reverse
++ lexicographic ordering with respect to the position that the variables
++ appear in the list of variables parameter.
HomogeneousDistributedMultivariatePolynomial(vl, R) : public == private where
  vl : List Symbol
  R  : Ring
  E   ==> HomogeneousDirectProduct(#vl, NonNegativeInteger)
  OV  ==> OrderedVariableList(vl)
  public == PolynomialCategory(R, E, OV) with
      reorder : (%, List Integer) -> %
        ++ reorder(p, perm) applies the permutation perm to the variables
        ++ in a polynomial and returns the new correctly ordered polynomial
  private ==
    GeneralDistributedMultivariatePolynomial(vl, R, E)

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