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PolynomialIdeals

 UnivariatePolynomial
 compute the smallest univariate polynomial in an ideal

 Calling Sequence UnivariatePolynomial(v, J, X)

Parameters

 v - variable name J - polynomial ideal or a list or set of generator polynomials X - (optional) set of variable names

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{3}-{y}^{2},y-x⟩$
 ${J}{≔}⟨{y}{-}{x}{,}{{x}}^{{3}}{-}{{y}}^{{2}}⟩$ (1)
 > $\mathrm{UnivariatePolynomial}\left(x,J\right)$
 ${{x}}^{{3}}{-}{{x}}^{{2}}$ (2)
 > $K≔⟨{x}^{3}-{y}^{3}+1,{y}^{2}+2,12z{t}^{2}-2{t}^{3}+1⟩$
 ${K}{≔}⟨{{y}}^{{2}}{+}{2}{,}{-}{2}{}{{t}}^{{3}}{+}{12}{}{{t}}^{{2}}{}{z}{+}{1}{,}{{x}}^{{3}}{-}{{y}}^{{3}}{+}{1}⟩$ (3)
 > $\mathrm{UnivariatePolynomial}\left(x,K\right)$
 ${{x}}^{{6}}{+}{2}{}{{x}}^{{3}}{+}{9}$ (4)
 > $\mathrm{UnivariatePolynomial}\left(t,K\right)$
 ${0}$ (5)
 > $\mathrm{UnivariatePolynomial}\left(t,K,\left\{x,y,t\right\}\right)$
 ${2}{}{{t}}^{{3}}{-}{12}{}{{t}}^{{2}}{}{z}{-}{1}$ (6)
 > $\mathrm{IsZeroDimensional}\left(K,\left\{x,y,t\right\}\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{alias}\left(\mathrm{α}=\mathrm{RootOf}\left({Z}^{3}+Z+1\right),\mathrm{β}=\mathrm{RootOf}\left({Z}^{5}+{Z}^{4}+2Z+3\right)\right)$
 ${\mathrm{α}}{,}{\mathrm{β}}$ (8)
 > $L≔⟨6{x}^{2}\mathrm{β}+7{y}^{2}\mathrm{α}+3{x}^{4},-4{y}^{2}+4{x}^{2}{y}^{2}-6y{\mathrm{α}}^{3}⟩$
 ${L}{≔}⟨{3}{}{{x}}^{{4}}{+}{7}{}{{y}}^{{2}}{}{\mathrm{α}}{+}{6}{}{{x}}^{{2}}{}{\mathrm{β}}{,}{-}{3}{}{y}{}{{\mathrm{α}}}^{{3}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{-}{2}{}{{y}}^{{2}}⟩$ (9)
 > $\mathrm{UnivariatePolynomial}\left(x,L\right)$
 ${4}{}{{x}}^{{12}}{+}{16}{}{{x}}^{{10}}{}{\mathrm{β}}{+}{16}{}{{x}}^{{8}}{}{{\mathrm{β}}}^{{2}}{-}{8}{}{{x}}^{{10}}{-}{32}{}{{x}}^{{8}}{}{\mathrm{β}}{-}{32}{}{{x}}^{{6}}{}{{\mathrm{β}}}^{{2}}{+}{4}{}{{x}}^{{8}}{+}{16}{}{{x}}^{{6}}{}{\mathrm{β}}{+}{42}{}{{x}}^{{4}}{}{{\mathrm{α}}}^{{2}}{+}{16}{}{{x}}^{{4}}{}{{\mathrm{β}}}^{{2}}{+}{84}{}{{x}}^{{2}}{}{{\mathrm{α}}}^{{2}}{}{\mathrm{β}}{-}{21}{}{{x}}^{{4}}{-}{42}{}{{x}}^{{2}}{}{\mathrm{β}}$ (10)
 > $\mathrm{UnivariatePolynomial}\left(y,L\right)$
 ${-}{24}{}{{\mathrm{α}}}^{{2}}{}{\mathrm{β}}{}{{y}}^{{3}}{+}{36}{}{{\mathrm{α}}}^{{2}}{}{\mathrm{β}}{}{{y}}^{{2}}{-}{12}{}{{\mathrm{α}}}^{{2}}{}{{y}}^{{3}}{+}{28}{}{{y}}^{{5}}{+}{36}{}{{\mathrm{α}}}^{{2}}{}{{y}}^{{2}}{-}{24}{}{\mathrm{β}}{}{{y}}^{{3}}{-}{27}{}{{\mathrm{α}}}^{{2}}{}{y}{-}{12}{}{{y}}^{{3}}{+}{27}{}{\mathrm{α}}{}{y}{+}{27}{}{y}$ (11)
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