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It seems that the polynomial f itself does not contain the good information. Maybe 500 bits of precision is not enough?

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return points(L)

I get:

sage: G = plot_roots_of_f(1,20,.05) + plot_np

It seems that the polynomial f itself does not contain the good information. Maybe 500 bits of precision is not enough?information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return points(L)

I get:

sage: G = plot_roots_of_f(1,20,.05) + plot_np

sage: R
Multivariate Polynomial Ring in x, y over Rational Field

It seems that the polynomial f itself does not contain the good information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return points(L)
point(L)    # I am using `point` here because the sage function `points` is a list in your code

I get:

sage: plot_roots_of_f(1,20,.05) + plot_np

Replacing u and v by x and y to use the ring over the rational field seems to give the same thing.

sage: R
Multivariate Polynomial Ring in x, y over Rational Field

Are you sure about that polynomial f ?

It seems that the polynomial f itself does not contain the good information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return point(L)    # I am using `point` here because the sage function `points` is a list in your the code

I get:

sage: plot_roots_of_f(1,20,.05) + plot_np

Replacing u and v by x and y to use the ring over the rational field seems to give the same thing.

sage: R
Multivariate Polynomial Ring in x, y over Rational Field

Are you sure about that polynomial f ?

It seems that the polynomial f itself does not contain the good information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return point(L)  # using `point` because the sage function `points` is a list in the code

I get:

sage: plot_roots_of_f(1,20,.05) + plot_np

Replacing u and v by x and y to use the ring over the rational field seems to give the same thing.

sage: R
Multivariate Polynomial Ring in x, y over Rational Field

Are you sure about that polynomial f ?

It seems that the polynomial f itself does not contain the good information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return point(L) # using `point` because the sage function `points` is a list in the code

I get:

sage: plot_roots_of_f(1,20,.05) + plot_np

Replacing u and v by x and y to use the ring over the rational field seems to give the same thing.

sage: R
Multivariate Polynomial Ring in x, y over Rational Field

Are you sure about that polynomial f ?

sage: f.subs(u=20)
0

It seems that the polynomial f itself does not contain the good information.

sage: f.parent()
Multivariate Polynomial Ring in u, v over Real Field with 500 bits of precision

Because when I draw its roots with:

def plot_roots_of_f(start,stop,step):
roots_of_f(start,stop,step):
    u_range = srange(start, stop, step)
    L = [(u,v) for u in u_range 
               for v in f.subs(u=u).univariate_polynomial().roots(multiplicities=False) 
               if (u,v) in P]
    return point(L) # using `point` because the sage function `points` L

I get (points is a list in the code

code and overwrites the sage function points, so I get:

sage: plot_roots_of_f(1,20,.05) am using point below):
 
sage: point(roots_of_f(1,20,.05)) + plot_np
 
 
Replacing u and v by x and y to use the ring over the rational field seems to give the same thing.
 
sage: R
Multivariate Polynomial Ring in x, y over Rational Field
 
Are you sure about that polynomial f ? because the following seems weird:
 
sage: f.subs(u=20)
0