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PolynomialIdeals

 Operators
 binary operators for ideals

 Calling Sequence J + K J * K J / K J ^ n f in J J subset K Simplify(J)

Parameters

 J, K - polynomial ideals n - positive integer f - polynomial

Description

 • The Operators subpackage provides access to Add, Multiply, and Quotient as binary operators.  An exponentiation operator, which uses binary powering on Multiply, is also provided.  These operators are intended for interactive use with small examples.
 Note: The in and subset operators are bound to the IdealMembership and IdealContainment routines when the PolynomialIdeals package is loaded. They are not part of the Operators subpackage.
 • The arithmetic operators accept ordinary expressions as well as ideals.  Wherever an expression f is encountered, the operators construct  in an appropriate polynomial ring.
 • Unlike their respective commands in PolynomialIdeals, the arithmetic operators simplify their results to a canonical form using reduced Groebner bases. The Simplify command is also rebound so that it simplifies ideals to this same canonical form.  The parent PolynomialIdeals package command behavior of Simplify can still be accessed through the long form PolynomialIdeals[Simplify].
 • Operator overloading and the simplification of ideals to a canonical form is often very expensive and may be impractical for problems of even a moderate size.  In this case, don't use these operators.  Use the Add, Multiply, and Quotient commands in the parent PolynomialIdeals package, and apply the Simplify command selectively.
 • These operators are part of the Operators subpackage, and can be used in their binary form only after executing with(PolynomialIdeals[Operators]), or inside a use statement.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $\mathrm{with}\left(\mathrm{Operators}\right)$
 $\left[{\mathrm{*}}{,}{\mathrm{+}}{,}{\mathrm{Simplify}}{,}{\mathrm{^}}\right]$ (1)
 > $J≔⟨{x}^{2}-1⟩$
 ${J}{≔}⟨{{x}}^{{2}}{-}{1}⟩$ (2)
 > $K≔⟨{y}^{2}-y⟩$
 ${K}{≔}⟨{{y}}^{{2}}{-}{y}⟩$ (3)
 > $L≔⟨x+1,y-1⟩$
 ${L}{≔}⟨{x}{+}{1}{,}{y}{-}{1}⟩$ (4)
 > $J+L$
 $⟨{x}{+}{1}{,}{y}{-}{1}⟩$ (5)
 > $J+\frac{K}{L}$
 $⟨{{x}}^{{2}}{-}{1}{,}{{y}}^{{2}}{-}{y}⟩$ (6)
 > $\frac{J+K}{L}$
 $⟨{{x}}^{{2}}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{x}{}{y}{-}{y}⟩$ (7)
 > $\mathrm{L2}≔{L}^{2}$
 ${\mathrm{L2}}{≔}⟨{{x}}^{{2}}{+}{2}{}{x}{+}{1}{,}{{y}}^{{2}}{-}{2}{}{y}{+}{1}{,}{x}{}{y}{-}{x}{+}{y}{-}{1}⟩$ (8)
 > $f≔{x}^{2}+2x+1$
 ${f}{≔}{{x}}^{{2}}{+}{2}{}{x}{+}{1}$ (9)
 > $g≔y-1$
 ${g}{≔}{y}{-}{1}$ (10)
 > $f\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{L2}$
 ${\mathrm{true}}$ (11)
 > $g\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{L2}$
 ${\mathrm{false}}$ (12)
 > $\mathrm{L2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{subset}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}⟨f,g⟩$
 ${\mathrm{true}}$ (13)
 > $H≔⟨{y}^{2}-1,\mathrm{characteristic}=3⟩$
 ${H}{≔}⟨{{y}}^{{2}}{+}{2}⟩$ (14)
 > $\frac{H}{g}$
 $⟨{y}{+}{1}⟩$ (15)
 > $\mathrm{IdealInfo}\left[\mathrm{Characteristic}\right]\left(\right)$
 ${3}$ (16)
 > $\frac{g}{H}$
 $⟨{1}⟩$ (17)
 > $\mathrm{IdealInfo}\left[\mathrm{Characteristic}\right]\left(\right)$
 ${3}$ (18)