)set messages interponly off

Sturm(p:POLY(FRAC(INT))):List(POLY(FRAC(INT)))==
  local V,x,S
  if p=0 then error("The polynomial must not be 0")
  else if totalDegree(p)=0 then [p]
  else  
    V:=variables(p)
    if #(V)>1 then error("The polynomial must be univariate")
    x:=V.1
    q:=p/gcd(p,D(p,x))
    q1:???:=q
    q2:???:=D(q,x)
    S:=[q1,q2]
    repeat
      ???
      ...
      ???
    S

--Input:
--  L.....a list of rationals.
--Output:
--  The number of sign variations in L.
variation(L:List(FRAC(INT))):NNI==
  local v
  if L=[] then 0
  else
    v:=0
    p:=L.1
    for n in L repeat
      if n~=0 then
        if p~=0 and sign(p)~=sign(n) then v:=v+1
        p:=n
    v

--Input:
--  p.......a polynomial with rational coefficients.
--  a,b.....rational numbers.
--Output:
--  The number of real roots of p in [a,b).
--Checks:
--  If p is not univariate or a>b then an error is signalled by the code.
rootsCount(p:POLY(FRAC(INT)),a:FRAC(INT),b:FRAC(INT)):Union(NNI,String)==
  local V,x,S
  if a>b then
    error("The first endpoint of the interval cannot be strictly larger than the second.")
  V:=variables(p)
  if #(V)>1 then error("The polynomial must be univariate")
  else if p=0 then "infinite"
  else
    if #(V)=1 then x:=V.1
    S:=Sturm(p)
    variation(eval(S,x=a))-variation(eval(S,x=b))
    
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-- Put the rest of your code below here.
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