If two polynomials $s\left(x\right)$ and $t\left(x\right)$, with both $s\left(0\right)$ and $t\left(0\right)$ nonzero, are given, the QDispersion(s,t,q,x,'maximal') calling sequence returns their q-dispersion, that is, $\mathrm{qdis}\left(s\left(x\right),t\left(x\right)\right)=\max \{r\|r\mathrm{in}Z,r>=0,\mathrm{deg}\left(\gcd \left(s\left(q^{r}x\right),t\left(x\right)\right)\right)>=1\}$ if the option 'maximal' is specified. Otherwise, the QDispersion(s,t,q,x) calling sequence returns the set of all non-negative integers $r$ used in the definition of the q-dispersion.

If $s\left(x\right)$ and $t\left(x\right)$ are as above and $k,l$ are non-negative integers, then QDispersion(x^k*s,x^l*t,q,x) returns the same result as QDispersion(s,t,q,x), and similarly if the option 'maximal' is specified.

The efficient algorithm for computing the dispersion of two polynomials $\mathrm{dis}\left(s\left(x\right),t\left(x\right)\right)=\max \{r\|r\mathrm{in}Z,r>=0,\mathrm{deg}\left(\gcd \left(s\left(x+r\right),t\left(x\right)\right)\right)>=1\}$ is the algorithm by Yiu-Kwong Man and F.J.Wright. This algorithm is based on the factorization of the polynomials involved rather than on the resultant calculation as it was in earlier implementations. This algorithm is adapted for computing the q-dispersion of two polynomials.

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

$\mathrm{p1}≔\left(x^2+3\right){\left(x+q\right)}^2\;$$\mathrm{p2}≔\left(qx+1\right)\left(q^{100}+q^5{x}^2+3\right)\left(q^{20}x+1\right)$

$\mathrm{p1}≔\left(x^2+3\right){\left(x+q\right)}^2$

$\mathrm{p2}≔\left(qx+1\right)\left(q^{100}+q^5{x}^2+3\right)\left(q^{20}x+1\right)$

$\mathrm{QDispersion}\left(\mathrm{p1}\,\mathrm{p2}\,q\,x\right)$

$\left\{2\,21\right\}$

$\mathrm{QDispersion}\left(\mathrm{p1}\,\mathrm{p2}\,q\,x\,'\mathrm{maximal}'\right)$

$21$

$q≔10\;$$\mathrm{QDispersion}\left(\mathrm{p1}\,\mathrm{p2}\,q\,x\right)$

$q≔10$

$\left\{2\,21\right\}$

Khmelnov, D.E. "Improved Algorithms for Solving Difference and q-Difference Equations." Programming and Computer Software. Vol. 26 No. 2. (2000): 107-115. Translated from Programmirovanie. No. 2.

Man, Yiu-Kwong, and Wright, Francis J. "Fast Polynomial Dispersion Computation and its Application to Indefinite Summation." Proceedings of ISSAC'94, pp. 175-180. ACM Press: New York, 1994.