dispersion computes the set of non-negative integers i such that $\gcd \left(\mathrm{LREtools}\left[\mathrm{shift}\right]\left(\mathrm{p1}\,n\,i\right)\,\mathrm{p2}\right)\ne 1$.  If there are no such integers, the function returns FAIL. Effectively, the dispersion measures the integers that can be added to the indeterminate in p1 and get a polynomial that has a common factor with p2.

If any of the polynomials contain parameters, then the returned answer is the generic dispersion, in other words the dispersion of the polynomials obtained by replacing the parameters by random numbers.  As such, the answer returned will be subject to specialization problems.

The optional argument 'maximal' can be used to indicate that only the maximal dispersion is wanted.  Returns FAIL if there are no such integers.

autodispersion computes dispersion(p1, subs(var=var-1, p1), n). This function is provided as this quantity can be computed more efficiently than the equivalent call to dispersion.

This notion originated in the works of Abramov.  The algorithm used is based on the work of Yiu-Kwong Man and Francis J. Wright. (See References).

$\mathrm{with}\left(\mathrm{LREtools}\right)\:$

$\mathrm{dispersion}\left(x\,x+4\,x\right)$

$\left\{4\right\}$

$\mathrm{dispersion}\left(x^2+2\,x-6\,x\right)$

$\mathrm{FAIL}$

$\mathrm{dispersion}\left(\left(x-1\right)\left(x-a\right)\,\left(x-7\right)\left(x^5-3\right)\,x\right)$

$\mathrm{FAIL}$

$\mathrm{dispersion}\left(\left(x-7\right)\left(x^5-3\right)\,\left(x-1\right)\left(x-a\right)\,x\right)$

$\left\{6\right\}$

$\mathrm{dispersion}\left(\left(x-7\right)\left(x^5-3\right)\,\left(x-1\right)\left(x-a\right)\,x\,'\mathrm{maximal}'\right)$

$6$

$\mathrm{dispersion}\left(x^{15}-1\,{\left(x+1\right)}^{15}-1\,x\right)$

$\left\{1\right\}$

Abramov, S.A..  "On the summation of rational functions."  USSR Comp. Math. Phys. 11.  (1971): 324-330.

Abramov, S.A..  "Rational solutions of linear differential and difference equations with polynomial coefficients."  USSR Comp. Math. Phys. 29.  (1989): 7-12.

Man, Yiu-Kwong, and Wright, Francis J.  "Fast  Polynomial Dispersion Computation and its Application to Indefinite Summation."  Proceedings of ISSAC '94:  175-180