The Coefficient(Poly, n) calling sequence returns the coefficient of the nth power of the noncommutative indeterminate in Poly.

The Coefficients(Poly) calling sequence returns the sequence of coefficients of Poly.

The Degree(Poly) calling sequence returns the degree of Poly with respect to the noncommutative indeterminate.

The LeadingCoefficient(Poly)] calling sequence returns the leading coefficient of Poly.

The LowDegree(Poly) calling sequence returns the trailing degree of Poly.

The RandOrePoly(A) calling sequence returns a random Ore polynomial in the Ore algebra A.

The first argument A specifies the ring in which the polynomial is to be generated. The possible options are:

coeffs - Generate the coefficients

terms - Number of terms in the noncommutative indeterminate

degree - Degree on the noncommutative indeterminate

The TrailingCoefficient(Poly) calling sequence returns the trailing coefficient of A.

The VariableDegree(Poly, A) calling sequence returns the maximal degree of coefficients of Poly with respect to the variable in A.  Note that the coefficients of Poly are supposed to be polynomials in the variable.

For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

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

$\mathrm{with}\left(\mathrm{OreTools}\left[\mathrm{Utility}\right]\right)\:$

$\mathrm{Poly}≔'\mathrm{OrePoly}'\left(0\,2n-1\,0\,\frac{1}{n}\right)$

$\mathrm{Poly}≔\mathrm{OrePoly}\left(0\,2n-1\,0\,\frac{1}{n}\right)$

$\mathrm{Coefficient}\left(\mathrm{Poly}\,1\right)$

$2n-1$

$\mathrm{Coefficient}\left(\mathrm{Poly}\,6\right)$

$0$

$\mathrm{Coefficients}\left(\mathrm{Poly}\right)$

$0,2n-1,0,\frac{1}{n}$

$\mathrm{Degree}\left(\mathrm{Poly}\right)$

$3$

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

$-\mathrm{\infty }$

$\mathrm{LeadingCoefficient}\left(\mathrm{Poly}\right)$

$\frac{1}{n}$

$\mathrm{LowDegree}\left(\mathrm{Poly}\right)$

$1$

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

$\mathrm{\infty }$

$\mathrm{TrailingCoefficient}\left(\mathrm{Poly}\right)$

$2n-1$

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

$0$

$A≔\mathrm{SetOreRing}\left(n\,'\mathrm{shift}'\right)\:$

$\mathrm{Poly}≔\mathrm{RandOrePoly}\left(A\right)$

$\mathrm{Poly}≔\mathrm{OrePoly}\left(72n^5+37n^4-23n^3+87n^2+44n+29\,-50n^5+23n^4+75n^3-92n^2+6n+74\,-17n^5-75n^4-10n^3-7n^2-40n+42\,-10n^5+62n^4-82n^3+80n^2-44n+71\,-62n^4+97n^3-73n^2-4n-83\,-7n^5+22n^4-55n^3-94n^2+87n-56\right)$

$\mathrm{Degree}\left(\mathrm{Poly}\right)$

$5$

$\mathrm{VariableDegree}\left(\mathrm{Poly}\,A\right)$

$5$

$\mathrm{VariableDegree}\left('\mathrm{OrePoly}'\left(0\right)\,A\right)$

$-\mathrm{\infty }$

$B≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)\:$

$C≔\mathrm{RandOrePoly}\left(B\,\mathrm{coeffs}=\mathrm{polynom}\left(\mathrm{degree}=3\,\mathrm{terms}=2\right)\,\mathrm{terms}=2\,\mathrm{degree}=10\right)$

$C≔\mathrm{OrePoly}\left(0\,0\,0\,0\,40x^3-81x\,11+95x\right)$

$\mathrm{VariableDegree}\left(C\,B\right)$

$3$

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

$\mathrm{RandOrePoly}\left(F\,\mathrm{coeffs}=\mathrm{ratpoly}\left(\mathrm{degnum}=1\,\mathrm{degden}=2\,\mathrm{terms}=2\,\mathrm{degree}=5\right)\,\mathrm{terms}=3\,\mathrm{degree}=10\right)$

$\mathrm{OrePoly}\left(0\,\frac{-87q+47x-90}{-88-48q}\,0\,\frac{16q+30x-27}{72qx-96}\,0\,0\,0\,0\,\frac{-51q+77x+95}{-28qx+55q}\right)$