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 algfuntoalgeq
 find a polynomial equation satisfied by an algebraic function

 Calling Sequence algfuntoalgeq(expr, y(z), ini, typ)

Parameters

 expr - algebraic or radical function in z y - name; holonomic function name z - name; variable of the holonomic function y ini - (optional) set; specify initial conditions for the resulting polynomial equation typ - (optional) 'algebraic' or 'rational'; specify the type of coefficients for the polynomial equation

Description

 • The algfuntoalgeq(expr, y(z)) command returns a polynomial in y and z that has expr as a root. The polynomial is not necessarily minimal.
 • If ini is specified, it is assigned a set of initial conditions for the polynomial to specify which branch is indicated (if possible).
 • If typ is specified as 'rational', the coefficients of the polynomial are of type rational or type name. This is particularly useful for finding a polynomial for an algebraic number.  The default is 'rational'.
 If typ is specified as 'algebraic' the coefficients are algebraic numbers. This option can be used with algebraicsubs.

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{algfuntoalgeq}\left(a\mathrm{RootOf}\left({\mathrm{_Z}}^{5}+1\right){x}^{\frac{2}{3}},y\left(x\right)\right)$
 ${{a}}^{{15}}{}{{x}}^{{10}}{+}{{y}}^{{15}}$ (1)
 > $\mathrm{algfuntoalgeq}\left(a\mathrm{RootOf}\left({\mathrm{_Z}}^{5}+1\right){x}^{\frac{2}{3}},y\left(x\right),'\mathrm{algebraic}'\right)$
 ${{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{5}}{+}{1}\right)}^{{3}}{}{{a}}^{{3}}{}{{x}}^{{2}}{-}{{y}}^{{3}}$ (2)
 > $\mathrm{algfuntoalgeq}\left({5}^{\frac{1}{3}}+3{7}^{\frac{2}{3}},y\left(x\right)\right)$
 ${{y}}^{{9}}{-}{3984}{}{{y}}^{{6}}{+}{5112147}{}{{y}}^{{3}}{-}{2342039552}$ (3)
 > $\mathrm{algfuntoalgeq}\left(1+\mathrm{RootOf}\left({\mathrm{_Z}}^{3}+1\right)x,y\left(x\right),\mathrm{ini}\right)$
 ${-}{{x}}^{{3}}{-}{{y}}^{{3}}{+}{3}{}{{y}}^{{2}}{-}{3}{}{y}{+}{1}$ (4)
 > $\mathrm{ini}$
 $\left\{{y}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{1}\right){,}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({0}\right){=}{0}\right\}$ (5)