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I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7 on a field k=GF2(a) and 2 lists z1=[1,a,0,x4,x5,0,0], z2=[1,a,0,a,a+1,x6,x7]. When I run f(z1), I have a polynomial, but with f(z2), I receive FractionFieldElement. So I can call f(z1).constant_coefficient(), but I could not do the same with f(z2).

F2=GF(2); Rr.<x> = F2[]
k.<a> = F2.extension(x^2+x+1) # looks like bad syntax: the dot in "F." on the left side
v=3
o1=2
o2=2
n=v+o1+o2
R=PolynomialRing(k,["x%d" %i for i in range (1,n+1)])
x=R.gens()
Fc=[] # o1 polynomials in the first layer
for im in range(o1):  # what's o1? Sorry, when I pasted, some mistakes appeared but I didn't control them
     f=0
     for i in range (v): # what's v?
          for j in range (i,v+o1):
                f+=k.random_element()*x[i]*x[j]
     for i in range (n-o2):
              f+=k.random_element()*x[i]
         f+=k.random_element()
         Fc.append(f)
for im in range(o1,o1+o2):
      f=0
      for i in range (v+o1):
           for j in range (i,n):
                f+=k.random_element()*x[i]*x[j]
      for i in range (n):
            f+=k.random_element()*x[i]
      f+=k.random_element()
      Fc.append(f)
w=vector([k.random_element() for i in range (o1+o2)])    
Fn=[R.random_element() for i in range (o1)]
while True:
        y_rd=[k.random_element() for j in range (v)]
        y=y_rd+[x[jj] for jj in range (v,v+o1)]+[0 for jjj in range (v+o1,n)]
        bc=[]
        Colist=[]
        for i in range (o1):
             Fn[i]=Fc[i](y)
             cc=w[i]-(Fn[i].constant_coefficient())
             bc.append(cc)
        mbc=vector(bc)
        for i in range (o1):
             Colist.append(Fn[i].coefficient(x[j]) for j in range (v,v+o1))
        Mtx=matrix(Colist)
        if rank(Mtx.augment(mbc))==rank(Mtx):
                break
yb=Mtx.solve_right(mbc)
ybb=list(yb)
zz1=y_rd+ybb
print(y)
for i in range (o1+o2):
      print(Fc[i])
      print(Fc[i](y))
      print(type(Fc[i](y)))
print('-----------------------------------------')
yy=zz1+[x[i] for i in range (v+o1,n)]
print(yy)
for i in range (o1+o2):
      print(Fc[i])
      print(Fc[i](yy))
      print(type(Fc[i](yy)))

Many thanks!

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7

@John Palmieri: I have edited my question. I am sorry because of my silly faults.

@Max Alekseyev: thanks so much.

@Max Alekseyev: thanks so much. @John Palmieri: I have edited my question. I am sorry because of my silly faults.

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7

F2=GF(2) Rr.<x> = F2[] F. = F2.extension(x^2+x+1) k=F R=PolynomialRing(k,["x%d" %i

F2=GF(2) Rr.<x> = F2[] F. = F2.extension(x^2+x+1) k=F R=PolynomialRing(k,["x%d" %i for i in range

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7

I am so sorry, cause I can't attach a picture or post the whole my code in a comment.

I am so sorry, cause I can't attach a picture to or post the whole my code in a comment.

yy=zz1+[x[i] for i in range (v+o1,n)] print(yy) for i in range (o1+o2): print(Fc[i]) print(Fci) print(type(F

while True: y_rd=[k.random_element() for j in range (v)] y=y_rd+[x[jj] for jj in range (v,v+o1)]+[0 for jjj in r

`F2=GF(2) Rr.<x> = F2[] F. = F2.extension(x^2+x+1) k=F R=PolynomialRing(k,["x%d" %i for i in range (1,n+1)]) x=R.

yy=zz1+[x[i] for i in range (v+o1,n)] print(yy) for i in range (o1+o2): print(Fc[i]) print(Fci) print(type(F

while True: y_rd=[k.random_element() for j in range (v)] y=y_rd+[x[jj] for jj in range (v,v+o1)]+[0 for jjj in r

F2=GF(2) Rr.<x> = F2[] F. = F2.extension(x^2+x+1) k=F R=PolynomialRing(k,["x%d" %i for i in range (1,n+1)]) x=R.g

Why does a polynomial become an element of a fraction field? I have a multivariate polynomial f in x1,x2,x3,x4,x5,x6,x7