IsProper - Maple Help

Groebner

 IsProper
 decide if a given algebraic system is algebraically consistent

 Calling Sequence IsProper(J, X, characteristic=p)

Parameters

 J - a list or set of polynomials or a PolynomialIdeal X - (optional) a list or set of variables, a ShortMonomialOrder, or a MonomialOrder p - (optional) characteristic

Description

 • The IsProper command decides whether a set of polynomials J with respect to the indeterminates X is algebraically consistent (that is, whether J has at least one solution over the algebraic closure of the coefficient field). This is equivalent to testing whether 1 is a member of the ideal generated by J. The zero ideal is considered proper.
 • The variables of the system can be specified using an optional second argument X. If X is a ShortMonomialOrder then a Groebner basis of J with respect to X is computed. By default, X is the set of all indeterminates not appearing inside a RootOf command or radical when J is a list or set, or PolynomialIdeals[IdealInfo][Variables](J) if J is an ideal.
 • The optional argument characteristic=p specifies the ring characteristic when J is a list or set. This option has no effect when J is a PolynomialIdeal or when X is a MonomialOrder.
 • Note that the is_solvable command is deprecated.  It may not be supported in a future Maple release.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{2}-2xz+5,x{y}^{2}+y{z}^{3},3{y}^{2}-6{z}^{3}+1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{-}{2}{}{x}{}{z}{+}{5}{,}{y}{}{{z}}^{{3}}{+}{x}{}{{y}}^{{2}}{,}{-}{6}{}{{z}}^{{3}}{+}{3}{}{{y}}^{{2}}{+}{1}\right]$ (1)
 > $\mathrm{IsProper}\left(F\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsProper}\left(F,\mathrm{characteristic}=3\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{Basis}\left(F,\mathrm{tdeg}\left(x,y,z\right),\mathrm{characteristic}=3\right)$
 $\left[{1}\right]$ (4)
 > $\mathrm{IsProper}\left(F,\left\{x,y\right\}\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{Basis}\left(F,\mathrm{tdeg}\left(x,y\right)\right)$
 $\left[{1}\right]$ (6)
 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨F,x⟩$
 ${J}{≔}⟨{x}{,}{y}{}{{z}}^{{3}}{+}{x}{}{{y}}^{{2}}{,}{{x}}^{{2}}{-}{2}{}{x}{}{z}{+}{5}{,}{-}{6}{}{{z}}^{{3}}{+}{3}{}{{y}}^{{2}}{+}{1}⟩$ (7)
 > $\mathrm{IsProper}\left(J\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{Basis}\left(J,'\mathrm{tord}'\right)$
 $\left[{1}\right]$ (9)