#C7.txt
Help:=proc(): print(` ProbConts(), Wt(L,x) , GFp(LL,x) , Comp(n,k) , CompU(n,k) `):
print(`GuessMulPol1(L,x,k,d), GuessMulPol(L,x,k) `):
end:

#VD
ProbConts:=proc():
[[4,1,3],[5,3,2],[4,1,5],[4,1,5],
[4,1,5],[4,2,1],[2,5,1],[2,4,5],[6,5,1],
[2,4,6],[4,2,1],[2,1,6],[1,2,4],[1,4,3]]:
end:

#maps a list of integers to x[1]^L[1]*...x[nops(L)]^L[nops(L)]
Wt:=proc(L,x) local i:
mul(x[i]^L[i],i=1..nops(L)):
end:


#GFp(LL,x) 
GFp:=proc(LL,x) local i:
add(Wt(LL[i],x),i=1..nops(LL)):
end:
####end VD
#from C4.txt
#Comp(n,k): inputs non-neg. integers n and k and outputs the SET of
 # compositions of n into k parts (vectors of non-neg. integers that add-up to n)
 #of length k
 Comp:=proc(n,k) local S,a1,S1, s1:
 option remember:
 
 if k<0 or n<0 then
   RETURN({}):
 fi:


  if n=0 then
    RETURN({[0$k]}):
  fi:

   if k=0 then
     if n=0 then
       RETURN({[]}):
     else
       RETURN({}):
     fi:
   fi:

 S:={}:

  for a1 from 0 to n do
     S1:=Comp(n-a1,k-1):
     S:=S union {seq([a1,op(s1)], s1 in S1)}:
   od:
S:

end:
#end C4.txt
#CompU(n,k) the set of vectors with k non-neg. componets that add up to <=n
CompU:=proc(n,k) local n1:
{seq(op(Comp(n1,k)),n1=0..n)}:
end:

#GuessMulPol1(L,x,k,d): inputs a list L of data of the form [pt,value] where
#pt is a list of length k, and value is the value. Guesses a polynomial
#in x[1], ..., x[k] of degree d,P, such that P(L[i][1])=L[i][2]
GuessMulPol1:=proc(L,x,k,d) local S,a,s,P,i,var,eq,i1:

 if not (
type(L,list) and {seq(type(L[i],list),i=1..nops(L))}={true} 
 and {seq(nops(L[i]),i=1..nops(L))}={2}
 and  {seq(type(L[i][1],list),i=1..nops(L))}={true}
and  {seq(nops(L[i][1]),i=1..nops(L))}={k}) then
 print(`bad input`):
  RETURN(FAIL):
fi:
S:=CompU(d,k):

P:=add(a[s]*mul(x[i]^s[i],i=1..k), s in S):
var:={seq(a[s],s in S)}:

eq:={seq(subs({seq(x[i1]=L[i][1][i1],i1=1..k)},P)=L[i][2],i=1..nops(L))}:

if nops(eq)-nops(var)<=3 then
 print(`not enough data `):
  RETURN(FAIL):
fi:

 var:=solve(eq,var):

 if var=NULL then
    RETURN(FAIL):
  fi:

P:= subs(var,P):

if P=0 then
 RETURN(FAIL):
else
 RETURN(P):
fi:

 end:


#Added after class ended
#GuessMulPol(L,x,k): inputs a list L of data of the form [pt,value] where
#pt is a list of length k, and value is the value. Guesses a polynomial
#in x[1], ..., x[k] of degree <=Maxd ,P, such that P(L[i][1])=L[i][2]
GuessMulPol:=proc(L,x,k) local d,P,d1:

for d from 1  while nops(L)-nops(CompU(d,k))>3 do od:
d:=d-1:

for d1 from 1 to d do
 P:=GuessMulPol1(L,x,k,d1):
 if P<>FAIL then
  RETURN(P):
 fi:
od:

FAIL:

end: