HGDispersion - Maple Help

LREtools[HypergeometricTerm]

 HGDispersion
 return the hypergeometric dispersion of two polynomials depending on a hypergeometric term

 Calling Sequence HGDispersion(p, q, x, r)

Parameters

 p - first polynomial q - second polynomial x - independent variable, for example, x r - list of equations that specifies the tower of hypergeometric extensions

Description

 • The HGDispersion(p, q, x, r) command returns the hypergeometric dispersion of p and q, that is,

$\mathrm{D}=\mathrm{max}\left\{n\ge 0:\mathrm{deg}\left(\mathrm{gcd}\left(p,\left({E}^{n}\right)q\right)\right)>0\right\}$

where E: Ex=x+1 is the shift operator and $p\left(x\right)$ and $q\left(x\right)$ are polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each ${r}_{i}$ is specified by a hypergeometric term, that is, $\frac{{\mathrm{Er}}_{i}}{{r}_{i}}$ is a rational function over K. The HGDispersion function returns $-1$ if the hypergeometric dispersion is not defined.

 • The polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the formal parameter r. Each hypergeometric term in the list is specified by a name, for example, t. It can be specified directly in the form of an equation, for example, $t=n!$, or specified as a list consisting of the name of the term variable and the consecutive term ratio, for example, $\left[t,n+1\right]$.
 • The computation of hypergeometric dispersions is reduced to solving the $\mathrm{\sigma }$-orbit problem (see OrbitProblemSolution) in the shortened tower of hypergeometric extensions.

Examples

 > $\mathrm{with}\left(\mathrm{LREtools}\left[\mathrm{HypergeometricTerm}\right]\right):$
 > $\mathrm{alias}\left(\mathrm{\phi }=3+\frac{4\mathrm{RootOf}\left({x}^{2}+1\right)}{5}\right):$
 > $p≔{\mathrm{\phi }}^{4}{s}^{2}+{\mathrm{\phi }}^{2}s+1$
 ${p}{≔}{{\mathrm{\phi }}}^{{4}}{}{{s}}^{{2}}{+}{{\mathrm{\phi }}}^{{2}}{}{s}{+}{1}$ (1)
 > $q≔{s}^{2}+s+1$
 ${q}{≔}{{s}}^{{2}}{+}{s}{+}{1}$ (2)
 > $\mathrm{ext}≔\left[s={\mathrm{\phi }}^{x}\right]$
 ${\mathrm{ext}}{≔}\left[{s}{=}{{\mathrm{\phi }}}^{{x}}\right]$ (3)
 > $\mathrm{HGDispersion}\left(p,q,x,\mathrm{ext}\right)$
 ${2}$ (4)
 > $\mathrm{alias}\left(\mathrm{\phi }=\mathrm{RootOf}\left({x}^{3}-5\right)\right):$
 > $p≔{\mathrm{\phi }}^{4}{s}^{2}+{\mathrm{\phi }}^{2}s+1$
 ${p}{≔}{{\mathrm{\phi }}}^{{4}}{}{{s}}^{{2}}{+}{{\mathrm{\phi }}}^{{2}}{}{s}{+}{1}$ (5)
 > $q≔{s}^{2}+s+1$
 ${q}{≔}{{s}}^{{2}}{+}{s}{+}{1}$ (6)
 > $\mathrm{ext}≔\left[s={\mathrm{\phi }}^{x}\right]$
 ${\mathrm{ext}}{≔}\left[{s}{=}{{\mathrm{\phi }}}^{{x}}\right]$ (7)
 > $\mathrm{HGDispersion}\left(p,q,x,\mathrm{ext}\right)$
 ${-1}$ (8)
 > $p≔{2}^{4}{s}^{2}+{2}^{2}s+1+v$
 ${p}{≔}{16}{}{{s}}^{{2}}{+}{4}{}{s}{+}{v}{+}{1}$ (9)
 > $q≔16{\left(x+3\right)}^{2}{\left(x+2\right)}^{2}{\left(x+1\right)}^{2}{s}^{2}+4\left(x+3\right)\left(x+2\right)\left(x+1\right)s+1+8v$
 ${q}{≔}{16}{}{\left({x}{+}{3}\right)}^{{2}}{}{\left({x}{+}{2}\right)}^{{2}}{}{\left({x}{+}{1}\right)}^{{2}}{}{{s}}^{{2}}{+}{4}{}\left({x}{+}{3}\right){}\left({x}{+}{2}\right){}\left({x}{+}{1}\right){}{s}{+}{1}{+}{8}{}{v}$ (10)
 > $\mathrm{ext}≔\left[v={2}^{x},s=x!\right]$
 ${\mathrm{ext}}{≔}\left[{v}{=}{{2}}^{{x}}{,}{s}{=}{x}{!}\right]$ (11)
 > $\mathrm{HGDispersion}\left(sq,pv+s,x,\mathrm{ext}\right)$
 ${3}$ (12)

References

 Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.