The UnivariatePolynomial command computes a univariate polynomial in v of least degree that is contained in the ideal J. If no such polynomial exists, then zero is returned. A zero-dimensional ideal contains a univariate polynomial in every variable.

The first argument must be the variable in which a univariate polynomial is to be computed.  The second argument must be a polynomial ideal. An optional third argument overrides the default ring variables.

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

$J≔⟨x^3-y^2\,y-x⟩$

$J≔⟨y-x\,x^3-y^2⟩$

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

$x^3-x^2$

$K≔⟨x^3-y^3+1\,y^2+2\,12z{t}^2-2t^3+1⟩$

$K≔⟨y^2+2\,-2t^3+12z{t}^2+1\,x^3-y^3+1⟩$

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

$x^6+2x^3+9$

$\mathrm{UnivariatePolynomial}\left(t\,K\right)$

$0$

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

$2t^3-12z{t}^2-1$

$\mathrm{IsZeroDimensional}\left(K\,\left\{t\,x\,y\right\}\right)$

$\mathrm{true}$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(Z^3+Z+1\right)\,\mathrm{\beta }=\mathrm{RootOf}\left(Z^5+Z^4+2Z+3\right)\right)$

$\mathrm{\alpha },\mathrm{\beta }$

$L≔⟨6x^2\mathrm{\beta }+7y^2\mathrm{\alpha }+3x^4\,-4y^2+4x^2{y}^2-6y{\mathrm{\alpha }}^3⟩$

$L≔⟨3x^4+7y^2\mathrm{\alpha }+6x^2\mathrm{\beta }\,-3y{\mathrm{\alpha }}^3+2x^2{y}^2-2y^2⟩$

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

$4x^{12}+16x^{10}\mathrm{\beta }+16x^8{\mathrm{\beta }}^2-8x^{10}-32x^8\mathrm{\beta }-32x^6{\mathrm{\beta }}^2+4x^8+16x^6\mathrm{\beta }+42x^4{\mathrm{\alpha }}^2+16x^4{\mathrm{\beta }}^2+84x^2{\mathrm{\alpha }}^2\mathrm{\beta }-21x^4-42x^2\mathrm{\beta }$

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

$-24{\mathrm{\alpha }}^2\mathrm{\beta }{y}^3+36{\mathrm{\alpha }}^2\mathrm{\beta }{y}^2-12{\mathrm{\alpha }}^2{y}^3+28y^5+36{\mathrm{\alpha }}^2{y}^2-24\mathrm{\beta }{y}^3-27{\mathrm{\alpha }}^{2}y-12y^3+27\mathrm{\alpha }y+27y$