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You are not using it the correct way. You defined f(x,y) over the symbolic ring but then you are doing the computations f(a, b) over the finite field. So, at every step it has to coerce the integers to the finite field. You should be using the PolynomialRing to get your variables and constants coerced to the finite field at the very outset.

Try the following code. It is fast.

f(x,y) = y^3 + y + 4*x
print "Original polynomial's base ring:", f.base_ring()
k = GF(25, 'a')
F.<x,y> = PolynomialRing(k)  # x,y are not symbolic variables now
f = y^3 + y + 4*x
print "New base ring:", f.base_ring()
count = 0
for a in k:
    for b in k:
        if not f.subs(x=a, y=b):
            count += 1
print count

You are not using it the correct way. You defined f(x,y) over the symbolic ring but then you are doing the computations f(a, b) over the finite field. So, at every step it has to coerce the integers everything to the finite field. You should be using the PolynomialRing to get your variables and constants coerced to the finite field at the very outset.

Try the following code. It is fast.

f(x,y) = y^3 + y + 4*x
print "Original polynomial's base ring:", f.base_ring()
k = GF(25, 'a')
F.<x,y> = PolynomialRing(k)  # x,y are not symbolic variables now
f = y^3 + y + 4*x
print "New base ring:", f.base_ring()
count = 0
for a in k:
    for b in k:
        if not f.subs(x=a, y=b):
            count += 1
print count

You are not using it the correct way. You defined f(x,y) over the symbolic ring but then you are doing the computations f(a, b) over the finite field. So, at every step it has to coerce everything to the finite field. You should be using the PolynomialRing to get your variables and constants coerced to the finite field at the very outset.

Try the following code. It is fast.

f(x,y) = y^3 + y + 4*x
print "Original polynomial's base ring:", f.base_ring()
k = GF(25, 'a')
F.<x,y> = PolynomialRing(k)  # x,y are not symbolic variables now
f = y^3 + y + 4*x
print "New base ring:", f.base_ring()
count = 0
for a in k:
    for b in k:
        if not f.subs(x=a, y=b):
            count += 1
print count

Original polynomial's base ring: Symbolic Ring
New base ring: Finite Field in a of size 5^2
25