The IdealInfo subpackage is a collection of simple procedures that return information about polynomial ideals.  They are intended to serve as a programmer interface to the PolynomialIdeal data structure. For compatibility with future releases, it is strongly recommended that you use these commands instead of accessing the operands of the data structure directly.

The Generators, Characteristic, and Variables commands return the generators, ring characteristic, and ring variables, respectively.  The Generators command is also aliased as PolynomialIdeals:-Generators.

The Parameters command returns the set of indeterminates appearing in the generators that are not ring variables.  The NonVariables command returns the set of all indeterminates that appear inside a radical or RootOf.

The KnownGroebnerBases command outputs the set of monomial orders for which Groebner bases are stored.  An optional second argument restricts the orders to a certain set of variables. To obtain the actual Groebner bases, use the Groebner:-Basis command.

The DefaultMonomialOrder command returns a monomial order for which a Groebner basis is stored or can be computed quickly. Optional arguments can specify both the type of monomial order (for example, 'plex') and the set of variables to use.

IdealInfo is a subpackage of the PolynomialIdeals package, and its various commands can be used in the form IdealInfo:-command(arguments) only after executing with(PolynomialIdeals).  However, they can always be accessed through the long form of the command using PolynomialIdeals:-IdealInfo:-command(arguments).

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

$J≔⟨x^2-z\,y^2+2\mathrm{sqrt}\left(3-w\right)\,\left\{\mathrm{characteristic}=5\,\mathrm{variables}=\left\{x\,y\right\}\right\}⟩$

$J≔⟨y^2+2\sqrt{3+4w}\,x^2+4z⟩$

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

POLYNOMIALIDEAL(y^2+2*(3+4*w)^(1/2),x^2+4*z,variables = {x, y},characteristic = 5,known_groebner_bases = (table([])))

$\mathrm{Generators}\left(J\right)=\mathrm{IdealInfo}:-\mathrm{Generators}\left(J\right)$

$\left\{y^2+2\sqrt{3+4w}\,x^2+4z\right\}=\left\{y^2+2\sqrt{3+4w}\,x^2+4z\right\}$

$\mathrm{IdealInfo}:-\mathrm{Characteristic}\left(J\right)$

$5$

$\mathrm{IdealInfo}:-\mathrm{Variables}\left(J\right)$

$\left\{x\,y\right\}$

$\mathrm{IdealInfo}:-\mathrm{Parameters}\left(J\right)$

$\left\{w\,z\right\}$

$\mathrm{IdealInfo}:-\mathrm{NonVariables}\left(J\right)$

$\left\{w\right\}$

$\mathrm{IdealInfo}:-\mathrm{DefaultMonomialOrder}\left(J\right)$

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

$\mathrm{Groebner}:-\mathrm{Basis}\left(J\,\mathrm{plex}\left(x\,y\right)\right)$

$\left[y^2+2\sqrt{3+4w}\,x^2+4z\right]$

$\mathrm{Groebner}:-\mathrm{Basis}\left(J\,\mathrm{tdeg}\left(x\,y\right)\right)$

$\left[y^2+2\sqrt{3+4w}\,x^2+4z\right]$

$\mathrm{IdealInfo}:-\mathrm{KnownGroebnerBases}\left(J\right)$

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

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

POLYNOMIALIDEAL(y^2+2*(3+4*w)^(1/2),x^2+4*z,variables = {x, y},characteristic = 5,known_groebner_bases = (table([(tdeg(y,x))=[[1, x^2, x^2+4*z], [1, y^2, y^2+2*(3+4*w)^(1/2)]],(tdeg(x,y))=[[1, y^2, y^2+2*(3+4*w)^(1/2)], [1, x^2, x^2+4*z]],(plex(x,y))=[[1, y^2, y^2+2*(3+4*w)^(1/2)], [1, x^2, x^2+4*z]]])))