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Groebner

 UnivariatePolynomial
 compute a univariate polynomial

 Calling Sequence UnivariatePolynomial(v, J, X, characteristic=p)

Parameters

 v - variable J - a list or set of polynomials or a PolynomialIdeal X - (optional) list or set of variables of the system p - (optional) characteristic

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{3}-3xy,{x}^{2}y-2{y}^{2}+x\right]$
 ${F}{≔}\left[{{x}}^{{3}}{-}{3}{}{x}{}{y}{,}{{x}}^{{2}}{}{y}{-}{2}{}{{y}}^{{2}}{+}{x}\right]$ (1)
 > $\mathrm{UnivariatePolynomial}\left(x,F\right)$
 ${{x}}^{{5}}{+}{9}{}{{x}}^{{2}}$ (2)
 > $\mathrm{UnivariatePolynomial}\left(y,F\right)$
 ${{y}}^{{6}}{-}{3}{}{{y}}^{{3}}$ (3)
 > $\mathrm{UnivariatePolynomial}\left(y,F,\mathrm{characteristic}=3\right)$
 ${{y}}^{{6}}$ (4)

The ideal below has infinitely many solutions, yet a univariate polynomial in x exists.

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{4}+z{y}^{3},xz{y}^{3}+1,{z}^{2}{y}^{6}-{x}^{3}⟩$
 ${J}{≔}⟨{{x}}^{{4}}{+}{z}{}{{y}}^{{3}}{,}{x}{}{z}{}{{y}}^{{3}}{+}{1}{,}{{z}}^{{2}}{}{{y}}^{{6}}{-}{{x}}^{{3}}⟩$ (5)
 > $\mathrm{IsZeroDimensional}\left(J\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{NumberOfSolutions}\left(J\right)$
 ${\mathrm{\infty }}$ (7)
 > $\mathrm{UnivariatePolynomial}\left(x,J\right)$
 ${{x}}^{{5}}{-}{1}$ (8)

A univariate polynomial in y does not exist, however we can treat z as a parameter to obtain a univariate polynomial in y with coefficients in Q(z).

 > $\mathrm{UnivariatePolynomial}\left(y,J\right)$
 ${0}$ (9)
 > $\mathrm{UnivariatePolynomial}\left(y,J,\left\{x,y\right\}\right)$
 ${{z}}^{{5}}{}{{y}}^{{15}}{+}{1}$ (10)