'Compute the *dispersion set* of two polynomials.
For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as:
.. math::
\operatorname{J}(f, g)
& := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\
&  = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}
For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`.
Examples
>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x
Dispersion set and dispersion of a simple polynomial:
>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6
Note that the definition of the dispersion is not symmetric:
>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
>>> dispersion(gp, fp)
-oo
Computing the dispersion also works over field extensions:
>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain=\'QQ<sqrt(5)>\')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain=\'QQ<sqrt(5)>\')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]
We can even perform the computations for polynomials
having symbolic coefficients:
>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]
See Also
dispersion
References
1. [ManWright94]_
2. [Koepf98]_
3. [Abramov71]_
4. [Man93]_'

def dispersionset(f, g=None):

    from sympy.polys.dispersion import dispersionset
    return dispersionset(f, g)
