gfun[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 of the function
z - name of the variable associated with y
ini - (optional) initial conditions to specify a solution of the resulting polynomial equation
typ - (optional) the type of coefficients to use in the polynomial equation. Must be either the string 'rational' or 'algebraic'. The default is 'rational'.
Description
- This function returns a polynomial in
yandzthat hasexpras a root. The polynomial is not necessarily minimal. - When the parameter
iniis used, it will be assigned a set of initial conditions for the polynomial to specify which branch is meant, when possible. - When the parameter
typis 'rational', the coefficients of the polynomial will be of type rational and/or type name. This is particularly useful for finding a polynomial for an algebraic number. When thetypis 'algebraic' the coefficients will be algebraic numbers. This option can be used with algebricsubs .
Examples
> with(gfun):> f:=(1-sqrt(1-4*z))/2/z;\[\frac{1-\sqrt{1-4 z}}{2 z}\]
> algfuntoalgeq(f,y(z),'ini');\[y^{2} z -y +1\]
> ini;\[\{y \! \left(0\right) = 1, D\! \left(y \right)\! \left(0\right) = 1, D^{\left(2\right)}\! \left(y \right)\! \left(0\right) = 4\}\]
> f:=a*RootOf(_Z^5+1)*x^(2/3);\[a \textrm{RootOf}\left(\_Z^{5}+1\right) x^{{2}/{3}}\]
> algfuntoalgeq(f,y(x));\[a^{15} x^{10}+y^{15}\]
> algfuntoalgeq(f,y(x),'algebraic');\[\textrm{RootOf}\left(\_Z^{5}+1\right)^{3} a^{3} x^{2}-y^{3}\]
> algfuntoalgeq(5^(1/3)+3*7^(2/3),y(x));\[y^{9}-3984 y^{6}+5112147 y^{3}-2342039552\]