 PolynomialIdeals - Maple Programming Help

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PolynomialIdeals

 Simplify
 put an ideal into a canonical form

 Calling Sequence Simplify(J) Simplify(J, X) Simplify(J, T) Simplify(J, S) Simplify(J, Z)

Parameters

 J - polynomial ideal, or sequence of ideals X - (optional) set of variables T - (optional) monomial ordering S - (optional) set of substitutions Z - (optional) set of inequations

Description

 • The Simplify command attempts to simplify an ideal using a Groebner basis.  It always detects whether the ideal is equivalent to the entire polynomial ring.
 • The Simplify command can be applied directly to sequences of ideals, but must be mapped onto lists and sets.
 • A monomial order, T, can be specified as an optional argument, forcing ideals into a canonical form.
 • The set of ring variables can also be overridden using an optional argument.  If the new variables are a subset of the original ones, this computes the extension of the ideal in the new polynomial ring. The Contract command inverts this process when it is possible to do so.  Groebner bases that are stored by the system are not preserved under this operation.
 • Simplify can optionally perform syntactic substitutions prior to the computation of a Groebner basis.  This is equivalent to using the eval command on the ideal generators.  Stored Groebner bases are not preserved under this operation.
 • Simplify can also simplify under the assumption that certain expressions are not equal.  This functionality is equivalent to that of the Saturate command, and can be of great assistance on large problems.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{\left({x}^{4}-{x}^{2}+y\right)}^{6},{\left(x-y\right)}^{5},{y}^{2}-x⟩$
 ${J}{≔}⟨{\left({x}{-}{y}\right)}^{{5}}{,}{\left({{x}}^{{4}}{-}{{x}}^{{2}}{+}{y}\right)}^{{6}}{,}{{y}}^{{2}}{-}{x}⟩$ (1)
 > $\mathrm{Simplify}\left(J\right)$
 $⟨{{x}}^{{3}}{,}{y}{}{{x}}^{{2}}{,}{{y}}^{{2}}{-}{x}⟩$ (2)
 > $K≔⟨{\left({x}^{2}+{y}^{2}\right)}^{6},{\left(2y-zx\right)}^{2},{z}^{2}xy⟩$
 ${K}{≔}⟨{\left({{x}}^{{2}}{+}{{y}}^{{2}}\right)}^{{6}}{,}{\left({-}{z}{}{x}{+}{2}{}{y}\right)}^{{2}}{,}{{z}}^{{2}}{}{x}{}{y}⟩$ (3)
 > $\mathrm{Simplify}\left(K,\mathrm{plex}\left(x,y\right)\right)$
 $⟨{{y}}^{{3}}{,}{x}{}{y}{,}{{z}}^{{2}}{}{{x}}^{{2}}{+}{4}{}{{y}}^{{2}}⟩$ (4)
 > $\mathrm{Simplify}\left(K,\mathrm{plex}\left(x,y\right),\left\{z=1\right\}\right)$
 $⟨{{y}}^{{3}}{,}{x}{}{y}{,}{{x}}^{{2}}{+}{4}{}{{y}}^{{2}}⟩$ (5)
 > $S≔\left\{⟨{x}^{2}-x+1,x-y⟩,⟨{y}^{2}-y+1,y-x⟩\right\}$
 ${S}{≔}\left\{⟨{x}{-}{y}{,}{{x}}^{{2}}{-}{x}{+}{1}⟩{,}⟨{y}{-}{x}{,}{{y}}^{{2}}{-}{y}{+}{1}⟩\right\}$ (6)
 > $\mathrm{map}\left(\mathrm{IdealInfo}:-\mathrm{DefaultMonomialOrder},S\right)$
 $\left\{{\mathrm{tdeg}}{}\left({x}{,}{y}\right){,}{\mathrm{tdeg}}{}\left({y}{,}{x}\right)\right\}$ (7)
 > $\mathrm{map}\left(\mathrm{Simplify},S\right)$
 $\left\{⟨{x}{-}{y}{,}{{y}}^{{2}}{-}{y}{+}{1}⟩{,}⟨{y}{-}{x}{,}{{x}}^{{2}}{-}{x}{+}{1}⟩\right\}$ (8)

Force a canonical form.

 > $\mathrm{map}\left(\mathrm{Simplify},S,\mathrm{tdeg}\left(x,y\right)\right)$
 $\left\{⟨{x}{-}{y}{,}{{y}}^{{2}}{-}{y}{+}{1}⟩\right\}$ (9)
 > $R≔⟨{z}^{2}+1-2y-2m+{y}^{2}+2ym-3{m}^{2},{x}^{2}+{y}^{2}-{m}^{2},1-2x-2m+{x}^{2}+2xm+{t}^{2},2-2z+{z}^{2}+{t}^{2}+2tm-2t-2m,5-6m-4z+{m}^{2}+4mz+{z}^{2}+4{t}^{2}-4t-4ty-4tm+2y+{y}^{2}+2ym⟩$
 ${R}{≔}⟨{-}{{m}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{,}{2}{}{x}{}{m}{+}{{t}}^{{2}}{+}{{x}}^{{2}}{-}{2}{}{m}{-}{2}{}{x}{+}{1}{,}{2}{}{t}{}{m}{+}{{t}}^{{2}}{+}{{z}}^{{2}}{-}{2}{}{m}{-}{2}{}{t}{-}{2}{}{z}{+}{2}{,}{-}{3}{}{{m}}^{{2}}{+}{2}{}{y}{}{m}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{-}{2}{}{m}{-}{2}{}{y}{+}{1}{,}{{m}}^{{2}}{-}{4}{}{t}{}{m}{+}{2}{}{y}{}{m}{+}{4}{}{m}{}{z}{+}{4}{}{{t}}^{{2}}{-}{4}{}{t}{}{y}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{-}{6}{}{m}{-}{4}{}{t}{+}{2}{}{y}{-}{4}{}{z}{+}{5}⟩$ (10)

R is a difficult system to solve.

 > $\mathrm{HilbertDimension}\left(R\right)$
 ${1}$ (11)
 > $\mathrm{R1}≔\mathrm{Simplify}\left(R,\left\{m\ne 0\right\}\right):$

R1 is much easier to solve.

 > $\mathrm{HilbertDimension}\left(\mathrm{R1}\right)$
 ${0}$ (12)