I must type the variable if I want the right result.

With theses types there is no problem :

(1) -> coefficient (numer (12 * (sin x)^3 * z), (sin x)::Kernel Expression Integer, 3)

(1)

But without type I get 0 :

coefficient (numer (12 * (sin x)^3 * z), sin z, 3)

   There are 2 exposed and 2 unexposed library operations named 
      coefficient having 3 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op coefficient
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      coefficient with argument type(s) 
      SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))
                             Expression(Integer)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

A simpler example is:

coefficient (numer((sin z)^2), sin z::Kernel EXPR INT, 2)

(2)

coefficient (numer((sin z)^2), sin z, 2)

   There are 2 exposed and 2 unexposed library operations named 
      coefficient having 3 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op coefficient
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      coefficient with argument type(s) 
      SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))
                             Expression(Integer)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

I modified POLYCAT as follows:

    coefficient(p,v,n) == 
         output(hconcat(["POLYCAT:", p::OutputForm, v::OutputForm, n::OutputForm]))$OutputPackage
         output(hconcat(["POLYCAT:", univariate(p,v)::OutputForm]))$OutputPackage
         coefficient(univariate(p,v),n)

)abbrev category POLYCAT PolynomialCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, monomial, coefficient, differentiate, eval
++ Related Constructors: Polynomial, DistributedMultivariatePolynomial
++ Also See: UnivariatePolynomialCategory
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category for general multi-variate polynomials over a ring
++ R, in variables from VarSet, with exponents from the
++ \spadtype{OrderedAbelianMonoidSup}.

PolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, VarSet:OrderedSet):
        Category ==
  Join(PartialDifferentialRing VarSet, FiniteAbelianMonoidRing(R, E),
       Evalable %, InnerEvalable(VarSet, R),
       InnerEvalable(VarSet, %), RetractableTo VarSet,
       FullyLinearlyExplicitRingOver R) with
    -- operations
    degree : (%,VarSet) -> NonNegativeInteger
      ++ degree(p,v) gives the degree of polynomial p with respect to the variable v.
    degree : (%,List(VarSet)) -> List(NonNegativeInteger)
      ++ degree(p,lv) gives the list of degrees of polynomial p
      ++ with respect to each of the variables in the list lv.
    coefficient: (%,VarSet,NonNegativeInteger) -> %
      ++ coefficient(p,v,n) views the polynomial p as a univariate
      ++ polynomial in v and returns the coefficient of the \spad{v**n} term.
    coefficient: (%,List VarSet,List NonNegativeInteger) -> %
      ++ coefficient(p, lv, ln) views the polynomial p as a polynomial
      ++ in the variables of lv and returns the coefficient of the term
      ++ \spad{lv**ln}, i.e. \spad{prod(lv_i ** ln_i)}.
    monomials: % -> List %
      ++ monomials(p) returns the list of non-zero monomials of polynomial p, i.e.
      ++ \spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.
    univariate   : (%,VarSet) -> SparseUnivariatePolynomial(%)
      ++ univariate(p,v) converts the multivariate polynomial p
      ++ into a univariate polynomial in v, whose coefficients are still
      ++ multivariate polynomials (in all the other variables).
    univariate   : % -> SparseUnivariatePolynomial(R)
        ++ univariate(p) converts the multivariate polynomial p,
        ++ which should actually involve only one variable,
        ++ into a univariate polynomial
        ++ in that variable, whose coefficients are in the ground ring.
        ++ Error: if polynomial is genuinely multivariate
    mainVariable  : % -> Union(VarSet,"failed")
           ++ mainVariable(p) returns the biggest variable which actually
           ++ occurs in the polynomial p, or "failed" if no variables are
           ++ present.
           ++ fails precisely if polynomial satisfies ground?
    minimumDegree : (%,VarSet) -> NonNegativeInteger
      ++ minimumDegree(p,v) gives the minimum degree of polynomial p
      ++ with respect to v, i.e. viewed a univariate polynomial in v
    minimumDegree : (%,List(VarSet)) -> List(NonNegativeInteger)
      ++ minimumDegree(p, lv) gives the list of minimum degrees of the
      ++ polynomial p with respect to each of the variables in the list lv
    monicDivide : (%,%,VarSet) -> Record(quotient:%,remainder:%)
       ++ monicDivide(a,b,v) divides the polynomial a by the polynomial b,
       ++ with each viewed as a univariate polynomial in v returning
       ++ both the quotient and remainder.
       ++ Error: if b is not monic with respect to v.
    monomial : (%,VarSet,NonNegativeInteger) -> %
        ++ monomial(a,x,n) creates the monomial \spad{a*x**n} where \spad{a} is
        ++ a polynomial, x is a variable and n is a nonnegative integer.
    monomial : (%,List VarSet,List NonNegativeInteger) -> %
        ++ monomial(a,[v1..vn],[e1..en]) returns \spad{a*prod(vi**ei)}.
    multivariate : (SparseUnivariatePolynomial(R),VarSet) -> %
        ++ multivariate(sup,v) converts an anonymous univariable
        ++ polynomial sup to a polynomial in the variable v.
    multivariate : (SparseUnivariatePolynomial(%),VarSet) -> %
        ++ multivariate(sup,v) converts an anonymous univariable
        ++ polynomial sup to a polynomial in the variable v.
    isPlus: % -> Union(List %, "failed")
        ++ isPlus(p) returns \spad{[m1,...,mn]} if polynomial \spad{p = m1 + ... + mn} and
        ++ \spad{n >= 2} and each mi is a nonzero monomial.
    isTimes: % -> Union(List %, "failed")
        ++ isTimes(p) returns \spad{[a1,...,an]} if polynomial \spad{p = a1 ... an}
        ++ and \spad{n >= 2}, and, for each i, ai is either a nontrivial constant in R or else of the
        ++ form \spad{x**e}, where \spad{e > 0} is an integer and x in a member of VarSet.
    isExpt: % -> Union(Record(var:VarSet, exponent:NonNegativeInteger),_
                       "failed")
        ++ isExpt(p) returns \spad{[x, n]} if polynomial p has the form \spad{x**n} and \spad{n > 0}.
    totalDegree : % -> NonNegativeInteger
        ++ totalDegree(p) returns the largest sum over all monomials
        ++ of all exponents of a monomial.
    totalDegree : (%,List VarSet) -> NonNegativeInteger
        ++ totalDegree(p, lv) returns the maximum sum (over all monomials of polynomial p)
        ++ of the variables in the list lv.
    variables : % -> List(VarSet)
        ++ variables(p) returns the list of those variables actually
        ++ appearing in the polynomial p.
    primitiveMonomials: % -> List %
        ++ primitiveMonomials(p) gives the list of monomials of the
        ++ polynomial p with their coefficients removed.
        ++ Note: \spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.
    if R has OrderedSet  then OrderedSet
    -- OrderedRing view removed to allow EXPR to define abs
    --if R has OrderedRing then OrderedRing
    if (R has ConvertibleTo InputForm) and
       (VarSet has ConvertibleTo InputForm) then
         ConvertibleTo InputForm
    if (R has ConvertibleTo Pattern Integer) and
       (VarSet has ConvertibleTo Pattern Integer) then
         ConvertibleTo Pattern Integer
    if (R has ConvertibleTo Pattern Float) and
       (VarSet has ConvertibleTo Pattern Float) then
         ConvertibleTo Pattern Float
    if (R has PatternMatchable Integer) and
       (VarSet has PatternMatchable Integer) then
         PatternMatchable Integer
    if (R has PatternMatchable Float) and
       (VarSet has PatternMatchable Float) then
         PatternMatchable Float
    if R has CommutativeRing then
      resultant : (%,%,VarSet) -> %
         ++ resultant(p,q,v) returns the resultant of the polynomials
         ++ p and q with respect to the variable v.
      discriminant : (%,VarSet) -> %
         ++ discriminant(p,v) returns the disriminant of the polynomial p
         ++ with respect to the variable v.
    if R has GcdDomain then
      GcdDomain
      content: (%,VarSet) -> %
        ++ content(p,v) is the gcd of the coefficients of the polynomial p
        ++ when p is viewed as a univariate polynomial with respect to the
        ++ variable v.
        ++ Thus, for polynomial 7*x**2*y + 14*x*y**2, the gcd of the
        ++ coefficients with respect to x is 7*y.
      primitivePart: % -> %
        ++ primitivePart(p) returns the unitCanonical associate of the
        ++ polynomial p with its content divided out.
      primitivePart: (%,VarSet) -> %
        ++ primitivePart(p,v) returns the unitCanonical associate of the
        ++ polynomial p with its content with respect to the variable v
        ++ divided out.
      squareFree: % -> Factored %
        ++ squareFree(p) returns the square free factorization of the
        ++ polynomial p.
      squareFreePart: % -> %
        ++ squareFreePart(p) returns product of all the irreducible factors
        ++ of polynomial p each taken with multiplicity one.

    -- assertions
    if R has canonicalUnitNormal then canonicalUnitNormal
             ++ we can choose a unique representative for each
             ++ associate class.
             ++ This normalization is chosen to be normalization of
             ++ leading coefficient (by default).
    if R has PolynomialFactorizationExplicit then
       PolynomialFactorizationExplicit
 add
    p:%
    v:VarSet
    ln:List NonNegativeInteger
    lv:List VarSet
    n:NonNegativeInteger
    pp,qq:SparseUnivariatePolynomial %
    eval(p:%, l:List Equation %) ==
      empty? l => p
      for e in l repeat
        retractIfCan(lhs e)@Union(VarSet,"failed") case "failed" => 
             error "cannot find a variable to evaluate"
      lvar:=[retract(lhs e)@VarSet for e in l]
      eval(p, lvar,[rhs e for e in l]$List(%))
    monomials p ==
--    zero? p => empty()
--    concat(leadingMonomial p, monomials reductum p)
--    replaced by sequential version for efficiency, by WMSIT, 7/30/90
      ml:= empty$List(%)
      while p ^= 0 repeat
        ml:=concat(leadingMonomial p, ml)
        p:= reductum p
      reverse ml
    isPlus p ==
      empty? rest(l := monomials p) => "failed"
      l
    isTimes p ==
      empty?(lv := variables p) or not monomial? p => "failed"
      l := [monomial(1, v, degree(p, v)) for v in lv]
--      one?(r := leadingCoefficient p) =>
      ((r := leadingCoefficient p) = 1) =>
        empty? rest lv => "failed"
        l
      concat(r::%, l)
    isExpt p ==
      (u := mainVariable p) case "failed" => "failed"
      p = monomial(1, u::VarSet, d := degree(p, u::VarSet)) =>
        [u::VarSet, d]
      "failed"
    -- coefficient(p,v,n) == coefficient(univariate(p,v),n)
    coefficient(p,v,n) == 
         output(hconcat(["POLYCAT:", p::OutputForm, v::OutputForm, n::OutputForm]))$OutputPackage
         output(hconcat(["POLYCAT:", univariate(p,v)::OutputForm]))$OutputPackage
         coefficient(univariate(p,v),n)
    coefficient(p,lv,ln) ==
       empty? lv =>
         empty? ln => p
         error "mismatched lists in coefficient"
       empty? ln  => error "mismatched lists in coefficient"
       coefficient(coefficient(univariate(p,first lv),first ln),
                   rest lv,rest ln)
    monomial(p,lv,ln) ==
       empty? lv =>
         empty? ln => p
         error "mismatched lists in monomial"
       empty? ln  => error "mismatched lists in monomial"
       monomial(monomial(p,first lv, first ln),rest lv, rest ln)
    retract(p:%):VarSet ==
      q := mainVariable(p)::VarSet
      q::% = p => q
      error "Polynomial is not a single variable"
    retractIfCan(p:%):Union(VarSet, "failed") ==
      ((q := mainVariable p) case VarSet) and (q::VarSet::% = p) => q
      "failed"
    mkPrim(p:%):% == monomial(1,degree p)
    primitiveMonomials p == [mkPrim q for q in monomials p]
    totalDegree p ==
        ground? p => 0
        u := univariate(p, mainVariable(p)::VarSet)
        d: NonNegativeInteger := 0
        while u ^= 0 repeat
          d := max(d, degree u + totalDegree leadingCoefficient u)
          u := reductum u
        d
    totalDegree(p,lv) ==
        ground? p => 0
        u := univariate(p, v:=(mainVariable(p)::VarSet))
        d: NonNegativeInteger := 0
        w: NonNegativeInteger := 0
        if member?(v, lv) then w:=1
        while u ^= 0 repeat
          d := max(d, w*(degree u) + totalDegree(leadingCoefficient u,lv))
          u := reductum u
        d

    if R has CommutativeRing then
        resultant(p1,p2,mvar) ==
          resultant(univariate(p1,mvar),univariate(p2,mvar))
        discriminant(p,var) ==
          discriminant(univariate(p,var))

    if R has IntegralDomain then
      allMonoms(l:List %):List(%) ==
        removeDuplicates_! concat [primitiveMonomials p for p in l]
      P2R(p:%, b:List E, n:NonNegativeInteger):Vector(R) ==
        w := new(n, 0)$Vector(R)
        for i in minIndex w .. maxIndex w for bj in b repeat
          qsetelt_!(w, i, coefficient(p, bj))
        w
      eq2R(l:List %, b:List E):Matrix(R) ==
        matrix [[coefficient(p, bj) for p in l] for bj in b]
      reducedSystem(m:Matrix %):Matrix(R) ==
        l := listOfLists m
        b := removeDuplicates_!
                           concat [allMonoms r for r in l]$List(List(%))
        d := [degree bj for bj in b]
        mm := eq2R(first l, d)
        l := rest l
        while not empty? l repeat
          mm := vertConcat(mm, eq2R(first l, d))
          l := rest l
        mm
      reducedSystem(m:Matrix %, v:Vector %):
       Record(mat:Matrix R, vec:Vector R) ==
        l := listOfLists m
        r := entries v
        b : List % := removeDuplicates_! concat(allMonoms r,
                          concat [allMonoms s for s in l]$List(List(%)))
        d := [degree bj for bj in b]
        n := #d
        mm := eq2R(first l, d)
        w := P2R(first r, d, n)
        l := rest l
        r := rest r
        while not empty? l repeat
          mm := vertConcat(mm, eq2R(first l, d))
          w := concat(w, P2R(first r, d, n))
          l := rest l
          r := rest r
        [mm, w]

    if R has PolynomialFactorizationExplicit then
       -- we might be in trouble if its actually only
       -- a univariate polynomial category - have to remember to
       -- over-ride these in UnivariatePolynomialCategory
       PFBR ==>PolynomialFactorizationByRecursion(R,E,VarSet,%)
       gcdPolynomial(pp,qq) ==
          gcdPolynomial(pp,qq)$GeneralPolynomialGcdPackage(E,VarSet,R,%)
       solveLinearPolynomialEquation(lpp,pp) ==
         solveLinearPolynomialEquationByRecursion(lpp,pp)$PFBR
       factorPolynomial(pp) ==
         factorByRecursion(pp)$PFBR
       factorSquareFreePolynomial(pp) ==
         factorSquareFreeByRecursion(pp)$PFBR
       factor p ==
         v:Union(VarSet,"failed"):=mainVariable p
         v case "failed" =>
           ansR:=factor leadingCoefficient p
           makeFR(unit(ansR)::%,
                  [[w.flg,w.fctr::%,w.xpnt] for w in factorList ansR])
         up:SparseUnivariatePolynomial %:=univariate(p,v)
         ansSUP:=factorByRecursion(up)$PFBR
         makeFR(multivariate(unit(ansSUP),v),
                [[ww.flg,multivariate(ww.fctr,v),ww.xpnt]
                 for ww in factorList ansSUP])
       if R has CharacteristicNonZero then
          mat: Matrix %
          conditionP mat ==
            ll:=listOfLists transpose mat   -- hence each list corresponds to a
                                            -- column, i.e. to one variable
            llR:List List R := [ empty() for z in first ll]
            monslist:List List % := empty()
            ch:=characteristic()$%
            for l in ll repeat
                mons:= "setUnion"/[primitiveMonomials u for u in l]
                redmons:List % :=[]
                for m in mons repeat
                    vars:=variables m
                    degs:=degree(m,vars)
                    deg1:List NonNegativeInteger
                    deg1:=[ ((nd:=d:Integer exquo ch:Integer)
                               case "failed" => return "failed" ;
                                nd::Integer::NonNegativeInteger)
                           for d in degs ]
                    redmons:=[monomial(1,vars,deg1),:redmons]
                    llR:=[[ground coefficient(u,vars,degs),:v] for u in l for v in llR]
                monslist:=[redmons,:monslist]
            ans:=conditionP transpose matrix llR
            ans case "failed" => "failed"
            i:NonNegativeInteger:=0
            [ +/[m*(ans.(i:=i+1))::% for m in mons ]
              for mons in monslist]

    if R has CharacteristicNonZero then
          charthRootlv: (%,List VarSet,NonNegativeInteger) -> Union(%,"failed")
          charthRoot p ==
            vars:= variables p
            empty? vars =>
              ans := charthRoot ground p
              ans case "failed" => "failed"
              ans::R::%
            ch:=characteristic()$%
            charthRootlv(p,vars,ch)
          charthRootlv(p,vars,ch) ==
            empty? vars =>
              ans := charthRoot ground p
              ans case "failed" => "failed"
              ans::R::%
            v:=first vars
            vars:=rest vars
            d:=degree(p,v)
            ans:% := 0
            while (d>0) repeat
               (dd:=(d::Integer exquo ch::Integer)) case "failed" =>
                      return "failed"
               cp:=coefficient(p,v,d)
               p:=p-monomial(cp,v,d)
               ansx:=charthRootlv(cp,vars,ch)
               ansx case "failed" => return "failed"
               d:=degree(p,v)
               ans:=ans+monomial(ansx,v,dd::Integer::NonNegativeInteger)
            ansx:=charthRootlv(p,vars,ch)
            ansx case "failed" => return "failed"
            return ans+ansx

    monicDivide(p1,p2,mvar) ==
       result:=monicDivide(univariate(p1,mvar),univariate(p2,mvar))
       [multivariate(result.quotient,mvar),
        multivariate(result.remainder,mvar)]

    if R has GcdDomain then
      if R has EuclideanDomain and R has CharacteristicZero then
       squareFree p == squareFree(p)$MultivariateSquareFree(E,VarSet,R,%)
      else
        squareFree p == squareFree(p)$PolynomialSquareFree(VarSet,E,R,%)
      squareFreePart p ==
        unit(s := squareFree p) * */[f.factor for f in factors s]
      content(p,v) == content univariate(p,v)
      primitivePart p ==
        unitNormal((p exquo content p) ::%).canonical
      primitivePart(p,v) ==
        unitNormal((p exquo content(p,v)) ::%).canonical
    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
      if (R has PatternMatchable Integer) and
         (VarSet has PatternMatchable Integer) then
           patternMatch(p:%, pat:Pattern Integer,
            l:PatternMatchResult(Integer, %)) ==
              patternMatch(p, pat,
                l)$PatternMatchPolynomialCategory(Integer,E,VarSet,R,%)
      if (R has PatternMatchable Float) and
         (VarSet has PatternMatchable Float) then
           patternMatch(p:%, pat:Pattern Float,
            l:PatternMatchResult(Float, %)) ==
              patternMatch(p, pat,
                l)$PatternMatchPolynomialCategory(Float,E,VarSet,R,%)

    if (R has ConvertibleTo Pattern Integer) and
       (VarSet has ConvertibleTo Pattern Integer) then
         convert(x:%):Pattern(Integer) ==
           map(convert, convert,
              x)$PolynomialCategoryLifting(E,VarSet,R,%,Pattern Integer)
    if (R has ConvertibleTo Pattern Float) and
       (VarSet has ConvertibleTo Pattern Float) then
         convert(x:%):Pattern(Float) ==
           map(convert, convert,
            x)$PolynomialCategoryLifting(E, VarSet, R, %, Pattern Float)
    if (R has ConvertibleTo InputForm) and
       (VarSet has ConvertibleTo InputForm) then
         convert(p:%):InputForm ==
           map(convert, convert,
                    p)$PolynomialCategoryLifting(E,VarSet,R,%,InputForm)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5427536097468235958-25px004.spad
      using old system compiler.
   POLYCAT abbreviates category PolynomialCategory 
******** Spad syntax error detected ********
The prior line was:

  171>       ml:= empty$List(%)

The current line is:

  172>       while p ^= 0 repeat

The number of valid tokens is 2.
The prior token was #S(TOKEN
                       :SYMBOL ^
                       :TYPE KEYWORD
                       :NONBLANK NIL
                       :LINE_NUM 172
                       :CHAR_NUM 14)
The current token is #S(TOKEN
                        :SYMBOL =
                        :TYPE KEYWORD
                        :NONBLANK NIL
                        :LINE_NUM 172
                        :CHAR_NUM 15)
The next token is #S(TOKEN
                     :SYMBOL 0
                     :TYPE NUMBER
                     :NONBLANK NIL
                     :LINE_NUM 172
                     :CHAR_NUM 17)

and obtained the following output:

coefficient(numer(sin x)^2, (sin x)::Kernel EXPR INT, 2)

(3)

coefficient(numer(sin x)^2, (sin x), 2)

   There are 2 exposed and 2 unexposed library operations named 
      coefficient having 3 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op coefficient
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      coefficient with argument type(s) 
      SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))
                             Expression(Integer)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Note also, that the result types are different.