The UnivariatePolynomial command returns a univariate polynomial in v of least degree in the ideal generated by J.  If no such polynomial exists then zero is returned. A zero-dimensional ideal contains a univariate polynomial for every variable.

An optional third argument X specifies the variables of the system. By default every indeterminate not appearing in a RootOf or radical is considered a variable when J is a list or a set. If J is a PolynomialIdeal a default set of variables is stored as part of the data structure.  See PolynomialIdeals[IdealInfo].

The optional argument characteristic=p specifies the ring characteristic when J is a list or a set. This option has no effect when J is a PolynomialIdeal, however you can specify J mod p as the first argument to obtain the desired result.

Note that the univpoly command is deprecated.  It may not be supported in a future Maple release.

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

$F≔\left[x^3-3xy\,x^{2}y-2y^2+x\right]$

$F≔\left[x^3-3xy\,x^{2}y-2y^2+x\right]$

$\mathrm{UnivariatePolynomial}\left(x\,F\right)$

$x^5+9x^2$

$\mathrm{UnivariatePolynomial}\left(y\,F\right)$

$y^6-3y^3$

$\mathrm{UnivariatePolynomial}\left(y\,F\,\mathrm{characteristic}=3\right)$

$y^6$

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

$J≔⟨x^4+z{y}^3\,xz{y}^3+1\,z^2{y}^6-x^3⟩$

$J≔⟨x^4+z{y}^3\,xz{y}^3+1\,z^2{y}^6-x^3⟩$

$\mathrm{IsZeroDimensional}\left(J\right)$

$\mathrm{false}$

$\mathrm{NumberOfSolutions}\left(J\right)$

$\mathrm{\infty }$

$\mathrm{UnivariatePolynomial}\left(x\,J\right)$

$x^5-1$

$\mathrm{UnivariatePolynomial}\left(y\,J\right)$

$0$

$\mathrm{UnivariatePolynomial}\left(y\,J\,\left\{x\,y\right\}\right)$

$z^5{y}^{15}+1$