list of polynomials in $\mathrm{Qx}$ with coefficients which are polynomials in $x$ over the field of rational functions in $q$

Let $k$ be a field of characteristic 0. Denote by $F$ the $q$-shift polynomial ring consisting of elements, each of which is a polynomial in $\mathrm{Qx}$, with coefficients which are polynomials in $x$ over $k\left(q\right)$. For a given operator $L\in F$, the Closure(L,Qx,x,q) calling sequence constructs a closure of $L$ in the $q$-shift polynomial ring $F$.

The output is a list of elements in $F$. Each element $R$ in this list represents a generator of the closure of $L$. For example, there exists $f\in k\left(q\right)\left[x\right]$ and $P\in F$, such that the torsion relation $P·L=f·R$ holds.

The Closure(L,Qx,x q,p) calling sequence constructs a local closure of $L$ at the irreducible(s) $p$. The output is a list of generators of the local closure of $L$.  For example, each element $R$ in the list is such that the torsion relation $P·L=g^i·R$ holds for some operator $P$, where $g\in p$ and $i\in \mathbb{N}$.

The parameter q does not have to be a variable. A constant value, such as $q=2$ is possible as well, including the case of a root of unity.

The optional argument func, if specified, is applied to the coefficients of the result with respect to $\mathrm{Qx}$; typical examples are expand or factor.

Note that setting infolevel[Closure]:=3 will cause some diagnostics to be printed during the computation.

'order'=o, where o is a monomial order

If this option is given, the Groebner[Basis] command, with respect to the given monomial order, will be applied to the computed closure.

'maximal'=truefalse (default: false)

This option only has an effect if a local closure is requested. If maximal=true (or maximal for short) is specified, then each element $R$ in the output list is such that the torsion relation $P·L={g\left(q^{j}x\right)}^i·R$ holds for some operator $P$, where $g\in p$, $i\in \mathbb{N}$, and $j\in \mathbb{Z}$. In other words, all generators for the $q$-shift-equivalence class(es) represented by $p$ are computed and returned.

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

$\left[\mathrm{AccurateQSummation}\,\mathrm{AreSameSolution}\,\mathrm{Closure}\,\mathrm{Desingularize}\,\mathrm{ExtendSeries}\,\mathrm{IsQHypergeometricTerm}\,\mathrm{IsSolution}\,\mathrm{PolynomialSolution}\,\mathrm{QBinomial}\,\mathrm{QBrackets}\,\mathrm{QDispersion}\,\mathrm{QECreate}\,\mathrm{QEfficientRepresentation}\,\mathrm{QFactorial}\,\mathrm{QGAMMA}\,\mathrm{QHypergeometricSolution}\,\mathrm{QMultiplicativeDecomposition}\,\mathrm{QPochhammer}\,\mathrm{QPolynomialNormalForm}\,\mathrm{QRationalCanonicalForm}\,\mathrm{QSimpComb}\,\mathrm{QSimplify}\,\mathrm{RationalSolution}\,\mathrm{RegularQPochhammerForm}\,\mathrm{SeriesSolution}\,\mathrm{UniversalDenominator}\,\mathrm{Zeilberger}\right]$

$L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)$

$L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)$

$C≔\mathrm{Closure}\left(L\,\mathrm{Qx}\,x\,q=\frac{1}{2}\right)$

$C≔\left[\left(\frac{1}{2}{x}^2-\frac{9}{2}x+9\right)\mathrm{Qx}+\frac{x^3}{128}-\frac{3x^2}{8}+\frac{45x}{8}-27\,\left(-\frac{3x}{4}+9\right){\mathrm{Qx}}^3+\left(-\frac{3}{1024}{x}^2-\frac{9}{128}x-\frac{405}{16}\right){\mathrm{Qx}}^2+\left(-\frac{9}{2048}{x}^2+\frac{81}{256}x-\frac{81}{16}\right)\mathrm{Qx}\,\left(\frac{8x}{63}-\frac{128}{21}\right){\mathrm{Qx}}^4+\left(\frac{1}{4032}{x}^2-\frac{5}{56}x+\frac{191}{7}\right){\mathrm{Qx}}^3+\left(-\frac{399x}{128}-\frac{63}{4}\right){\mathrm{Qx}}^2+\left(-\frac{51}{2048}{x}^2+\frac{243}{128}x-\frac{135}{4}\right)\mathrm{Qx}\,\left(-\frac{3x}{2}+9\right){\mathrm{Qx}}^2+\left(-\frac{3}{256}{x}^2-\frac{9}{64}x-\frac{405}{16}\right)\mathrm{Qx}-\frac{9x^2}{512}+\frac{81x}{128}-\frac{81}{16}\,\left(\frac{16x}{63}-\frac{128}{21}\right){\mathrm{Qx}}^3+\left(\frac{1}{1008}{x}^2-\frac{5}{28}x+\frac{191}{7}\right){\mathrm{Qx}}^2+\left(-\frac{399x}{64}-\frac{63}{4}\right)\mathrm{Qx}-\frac{51x^2}{512}+\frac{243x}{64}-\frac{135}{4}\right]$

$p≔-\frac{1}{3}{q}^{3}x+1$

$p≔-\frac{q^{3}x}{3}+1$

$C≔\mathrm{Closure}\left(L\,\mathrm{Qx}\,x\,q\,p\,\mathrm{factor}\right)$

$C≔\left[\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)\,\left(-3q^{2}x+9\right){\mathrm{Qx}}^3+\left(-3q^{10}{x}^2-9q^{7}x+18q^{6}x+9q^{5}x+9q^{4}x-27q^4-9q^{3}x+27q^3-27\right){\mathrm{Qx}}^2+9\left(q-1\right)\left(q^{4}x-3\right)\left(q^{3}x-3\right)q^3\mathrm{Qx}\,\frac{\left(q^{4}x-3\right){\mathrm{Qx}}^4}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{\left(q^{13}{x}^2+3q^{10}x-6q^{8}x-3q^{7}x+9q^7+9q^6-9q^5-18q^4-9q^3+9q^2+9q+9\right){\mathrm{Qx}}^3}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{9\left(q^2+q+1\right)\left(q^{6}x-q^{4}x-2q^{3}x+6q^3+3q^2-3\right){\mathrm{Qx}}^2}{q+1}-\frac{9\left(q^{4}x-3\right)\left(3q^{5}x+3q^{4}x+2q^{3}x-6q^2-9q-9\right)q^3\mathrm{Qx}}{q+1}\right]$

$C≔\mathrm{Closure}\left(L\,\mathrm{Qx}\,x\,q\,p\,'\mathrm{maximal}'\,\mathrm{factor}\right)$

$C≔\left[\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)\,\left(-3q^{2}x+9\right){\mathrm{Qx}}^3+\left(-3q^{10}{x}^2-9q^{7}x+18q^{6}x+9q^{5}x+9q^{4}x-27q^4-9q^{3}x+27q^3-27\right){\mathrm{Qx}}^2+9\left(q-1\right)\left(q^{4}x-3\right)\left(q^{3}x-3\right)q^3\mathrm{Qx}\,\frac{\left(q^{4}x-3\right){\mathrm{Qx}}^4}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{\left(q^{13}{x}^2+3q^{10}x-6q^{8}x-3q^{7}x+9q^7+9q^6-9q^5-18q^4-9q^3+9q^2+9q+9\right){\mathrm{Qx}}^3}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{9\left(q^2+q+1\right)\left(q^{6}x-q^{4}x-2q^{3}x+6q^3+3q^2-3\right){\mathrm{Qx}}^2}{q+1}-\frac{9\left(q^{4}x-3\right)\left(3q^{5}x+3q^{4}x+2q^{3}x-6q^2-9q-9\right)q^3\mathrm{Qx}}{q+1}\,\left(-3qx+9\right){\mathrm{Qx}}^2+\left(-3q^8{x}^2-9q^{6}x+18q^{5}x+9q^{4}x-27q^4+9q^{3}x+27q^3-9q^{2}x-27\right)\mathrm{Qx}+9\left(q-1\right)\left(q^{3}x-3\right)\left(q^{2}x-3\right)q^3\,\frac{\left(q^{3}x-3\right){\mathrm{Qx}}^3}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{\left(q^{11}{x}^2+3q^{9}x-6q^{7}x+9q^7-3q^{6}x+9q^6-9q^5-18q^4-9q^3+9q^2+9q+9\right){\mathrm{Qx}}^2}{q\left(q^2+q+1\right){\left(q-1\right)}^2{\left(q+1\right)}^2}+\frac{9\left(q^2+q+1\right)\left(q^{5}x-q^{3}x+6q^3-2q^{2}x+3q^2-3\right)\mathrm{Qx}}{q+1}-\frac{9\left(q^{3}x-3\right)\left(3q^{4}x+3q^{3}x+2q^{2}x-6q^2-9q-9\right)q^3}{q+1}\right]$

$A≔\mathrm{OreTools}:-\mathrm{SetOreRing}\left(\left[x\,q\right]\,'\mathrm{qshift}'\right)\:$

$L≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(L\,\mathrm{Qx}\right)$

$L≔\mathrm{OrePoly}\left({\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)\,\left(x-3\right)\left(qx-3\right)\right)$

$\mathrm{R2}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(C\left[2\right]\,\mathrm{Qx}\right)$

$\mathrm{R2}≔\mathrm{OrePoly}\left(0\,9\left(q-1\right)\left(q^{4}x-3\right)\left(q^{3}x-3\right)q^3\,-3q^{10}{x}^2-9q^{7}x+18q^{6}x+9q^{5}x+9q^{4}x-27q^4-9q^{3}x+27q^3-27\,-3q^{2}x+9\right)$

$\mathrm{OreTools}:-\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{R2}\,L\,A\,'\mathrm{P2}'\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{P2}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromOrePolyToPoly}\left(\mathrm{P2}\,\mathrm{Qx}\right)$

$\mathrm{P2}≔\frac{9q^3\left(q-1\right)\mathrm{Qx}}{q^{3}x-3}-\frac{3{\mathrm{Qx}}^2}{q^{3}x-3}$

$\mathrm{denom}\left(\mathrm{P2}\right)$

$q^{3}x-3$

$\mathrm{R4}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(C\left[4\right]\,\mathrm{Qx}\right)$

$\mathrm{R4}≔\mathrm{OrePoly}\left(9\left(q-1\right)\left(q^{3}x-3\right)\left(q^{2}x-3\right)q^3\,-3q^8{x}^2-9q^{6}x+18q^{5}x+9q^{4}x-27q^4+9q^{3}x+27q^3-9q^{2}x-27\,-3qx+9\right)$

$\mathrm{OreTools}:-\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{R4}\,L\,A\,'\mathrm{P4}'\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{P4}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromOrePolyToPoly}\left(\mathrm{P4}\,\mathrm{Qx}\right)$

$\mathrm{P4}≔\frac{9q^3\left(q-1\right)}{q^{2}x-3}-\frac{3\mathrm{Qx}}{q^{2}x-3}$

$\mathrm{denom}\left(\mathrm{P4}\right)$

$q^{2}x-3$

$\mathrm{numer}\left(\mathrm{eval}\left(p\,x=\frac{x}{q}\right)\right)$

$-q^{2}x+3$

$L≔{\left(x+2\right)}^2+\left(x+1\right){\left(x-2\right)}^2\mathrm{Qx}$

$L≔{\left(x+2\right)}^2+\left(x+1\right){\left(x-2\right)}^2\mathrm{Qx}$

$\mathrm{infolevel}\left[\mathrm{Closure}\right]≔3\:$

$C≔\mathrm{Closure}\left(L\,\mathrm{Qx}\,x\,-1\right)$

Closure:   "-1 is a 2 root of unity"
Closure:   "compute the matrix representation of the input operator"
Closure:   "the matrix representation is Matrix(2, 2, [[(x+2)^2,(x+1)*(x-2)^2],[(1-x)*(-x-2)^2*eta,(-x+2)^2]])"
Closure:   "compute the candidate primes and bounds for their multiplicities"
Closure:   "the candidate primes and bounds for their multiplicities are [[x-2, 2], [x+2, 2]]"
Closure:   "compute the local closures"
Closure:   "compute P such that P.L = (x-2)^j.R, 1<=j<=2"
Closure:   "compute P such that P.L = (x+2)^j.R, 1<=j<=2"

$C≔\left[{\left(x+2\right)}^2+\left(x+1\right){\left(x-2\right)}^2\mathrm{Qx}\&comma;\left(x^2-x-2\right){\mathrm{Qx}}^3+\left(-x^2-4x-4\right){\mathrm{Qx}}^2+\left(x-2\right)\mathrm{Qx}\&comma;\left(x^2-1\right){\mathrm{Qx}}^3+\mathrm{Qx}\&comma;\left(x^2-x-2\right){\mathrm{Qx}}^3+\left(-x^2-4x-4\right){\mathrm{Qx}}^2+\left(x-2\right)\mathrm{Qx}\&comma;\left(-x^2-x+2\right){\mathrm{Qx}}^2+\left(x^2-4x+4\right)\mathrm{Qx}+x+2\&comma;\left(1-x\right){\mathrm{Qx}}^2+\left(x^2-4x+4\right)\mathrm{Qx}+x+3\&comma;\left(-x^2-x+2\right){\mathrm{Qx}}^2+\left(x^2-4x+4\right)\mathrm{Qx}+x+2\right]$

The QDifferenceEquations[Closure] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.