 ParametricGCRD - Maple Help

OreTools

 ParametricGCRD
 determine the dependency of GCRD on a parameter Calling Sequence ParametricGCRD(P1, P2, ..., cond, Par, A) Parameters

 P1, P2, ... - Ore polynomials depending on a parameter; to define an Ore polynomial, use the OrePoly structure. cond - polynomial in Par Par - name; parameter A - Ore algebra; to define an Ore algebra, use the SetOreRing function. Description

 • The ParametricGCRD(P1, P2, ..., cond, Par, A) calling sequence finds the dependency of the greatest common right divisor (GCRD) of two or more Ore polynomials P1, P2, ... with polynomial coefficients and parametrized on Par that satisfies the polynomial condition $\mathrm{cond}\left(\mathrm{Par}\right)=0$.
 • The ParametricGCRD command returns an error message if the Ore polynomials P1, P2, ... do not have polynomial coefficients.
 • The return value depends on cond.
 1 If cond is a nonzero constant, the ParametricGCRD command returns NULL.
 2 If $\mathrm{cond}\ne 0$, the ParametricGCRD command returns $\mathrm{piecewise}\left(\mathrm{cond_1}=0,{g}_{1},...,\mathrm{cond_n}=0,{g}_{n}\right)$. For i=1, ... n, cond_i is a polynomial in Par, and g_i is an Ore polynomial with polynomial coefficients such that g_i=GCRD(P1, P2, ...) when $\mathrm{cond_i}\left(\mathrm{Par}\right)=0$. The product $\mathrm{cond_1}...\mathrm{cond_n}$ is equal to cond.
 3 If $\mathrm{cond}=0$, the ParametricGCRD command returns $\mathrm{piecewise}\left(\mathrm{cond_1}=0,{g}_{1},...,\mathrm{cond_n}=0,{g}_{n},\mathrm{g_otherwise}\right)$. For i=1, ... n, cond_i and g_i are defined as in the previous case and g_otherwise=GCRD(P1, P2, ...) for all other values of Par.
 4 If $\mathrm{cond}=0$ and there are no other GCRDs, the ParametricGCRD command returns the Ore polynomial that is the GCRD(P1, P2, ...) for all Par . Examples

 > $\mathrm{with}\left(\mathrm{OreTools}\right):$
 > $\mathrm{ERing}≔\mathrm{SetOreRing}\left(x,'\mathrm{shift}'\right)$
 ${\mathrm{ERing}}{≔}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{shift}}\right)$ (1)
 > $\mathrm{p1}≔\mathrm{OrePoly}\left(1,1,0,\left(a+2\right)x\right)$
 ${\mathrm{p1}}{≔}{\mathrm{OrePoly}}{}\left({1}{,}{1}{,}{0}{,}\left({a}{+}{2}\right){}{x}\right)$ (2)
 > $\mathrm{p2}≔\mathrm{OrePoly}\left(0,\left(a+2\right)\left(a+1\right)x\right)$
 ${\mathrm{p2}}{≔}{\mathrm{OrePoly}}{}\left({0}{,}\left({a}{+}{2}\right){}\left({a}{+}{1}\right){}{x}\right)$ (3)
 > $\mathrm{ParametricGCRD}\left(\mathrm{p1},\mathrm{p2},\left(a+1\right)\left(a+2\right)a,a,\mathrm{ERing}\right)$
 $\left\{\begin{array}{cc}{\mathrm{OrePoly}}{}\left({-1}\right)& {a}{=}{0}\\ {\mathrm{OrePoly}}{}\left({1}{,}{1}{,}{0}{,}{x}\right)& {a}{+}{1}{=}{0}\\ {\mathrm{OrePoly}}{}\left({1}{,}{1}\right)& {a}{+}{2}{=}{0}\end{array}\right\$ (4)
 > $\mathrm{DRing}≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right)$
 ${\mathrm{DRing}}{≔}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{differential}}\right)$ (5)
 > $\mathrm{p3}≔\mathrm{OrePoly}\left(a,-2a,x+a\right)$
 ${\mathrm{p3}}{≔}{\mathrm{OrePoly}}{}\left({a}{,}{-}{2}{}{a}{,}{x}{+}{a}\right)$ (6)
 > $\mathrm{p4}≔\mathrm{OrePoly}\left(ax,-2a\left(x-a+1\right),a{x}^{2}+x\right)$
 ${\mathrm{p4}}{≔}{\mathrm{OrePoly}}{}\left({a}{}{x}{,}{-}{2}{}{a}{}\left({x}{-}{a}{+}{1}\right){,}{a}{}{{x}}^{{2}}{+}{x}\right)$ (7)
 > $\mathrm{ParametricGCRD}\left(\mathrm{p3},\mathrm{p4},\left({a}^{2}-a\right)\left(a+1\right),a,\mathrm{DRing}\right)$
 $\left\{\begin{array}{cc}{\mathrm{OrePoly}}{}\left({a}{,}{-}{2}{}{a}{,}{a}{}{x}{+}{1}\right)& \left({a}{-}{1}\right){}{a}{=}{0}\\ {\mathrm{OrePoly}}{}\left({1}\right)& {a}{+}{1}{=}{0}\end{array}\right\$ (8)
 > $\mathrm{ParametricGCRD}\left(\mathrm{p3},\mathrm{p4},0,a,\mathrm{DRing}\right)$
 $\left\{\begin{array}{cc}{\mathrm{OrePoly}}{}\left({a}{,}{-}{2}{}{a}{,}{a}{}{x}{+}{1}\right)& \left({a}{-}{1}\right){}{a}{=}{0}\\ {\mathrm{OrePoly}}{}\left({1}\right)& {\mathrm{otherwise}}\end{array}\right\$ (9) References

 Glotov, P.E. "An algorithm of searching the greatest common divisor for Ore polynomial with polynomial coefficients depending on a parameter." Programming and Computer Software. Vol. 24 No. 6, (1998): 275-283.