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.

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

$F≔\left[x^2-2xz+5\,x{y}^2+y{z}^3\,3y^2-6z^3+1\right]$

$F≔\left[x^2-2xz+5\,y{z}^3+x{y}^2\,-6z^3+3y^2+1\right]$

$\mathrm{IsProper}\left(F\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{Basis}\left(F\,\mathrm{tdeg}\left(x\,y\,z\right)\,\mathrm{characteristic}=3\right)$

$\left[1\right]$

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

$\mathrm{false}$

$\mathrm{Basis}\left(F\,\mathrm{tdeg}\left(x\,y\right)\right)$

$\left[1\right]$

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

$J≔⟨F\,x⟩$

$J≔⟨x\,y{z}^3+x{y}^2\,x^2-2xz+5\,-6z^3+3y^2+1⟩$

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

$\mathrm{false}$

$\mathrm{Basis}\left(J\,'\mathrm{tord}'\right)$

$\left[1\right]$