EliminationIdeal - Maple Help

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PolynomialIdeals

 EliminationIdeal
 eliminate variables from an ideal (subring intersection)

 Calling Sequence EliminationIdeal(J, X)

Parameters

 J - polynomial ideal X - set of subring variable names

Description

 • The EliminationIdeal command eliminates variables from an ideal using a Groebner basis computation. The result of EliminationIdeal(J, X) is the intersection of the ideal J with the subring ${k}_{X}$.
 • Note: You cannot use the Intersect command to compute this result.  For any variables X, the polynomial ring ${k}_{X}$ is represented by the ideal $⟨1⟩$, and Intersect(J, <1>) = J.
 • The EliminationIdeal command can be used to perform nonlinear elimination on a general set of relations.  This is demonstrated below.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{2}-y,{y}^{2}+1⟩$
 ${J}{≔}⟨{{y}}^{{2}}{+}{1}{,}{{x}}^{{2}}{-}{y}⟩$ (1)
 > $\mathrm{EliminationIdeal}\left(J,\left\{x\right\}\right)$
 $⟨{{x}}^{{4}}{+}{1}⟩$ (2)
 > $\mathrm{EliminationIdeal}\left(J,\left\{y\right\}\right)$
 $⟨{{y}}^{{2}}{+}{1}⟩$ (3)
 > $K≔⟨x\left(1+{t}^{2}\right)-\left(1-{t}^{2}\right),y\left(1+{y}^{2}\right)-2t⟩$
 ${K}{≔}⟨{y}{}\left({{y}}^{{2}}{+}{1}\right){-}{2}{}{t}{,}{x}{}\left({{t}}^{{2}}{+}{1}\right){+}{{t}}^{{2}}{-}{1}⟩$ (4)
 > $\mathrm{EliminationIdeal}\left(K,\left\{x,y\right\}\right)$
 $⟨{x}{}{{y}}^{{6}}{+}{{y}}^{{6}}{+}{2}{}{x}{}{{y}}^{{4}}{+}{2}{}{{y}}^{{4}}{+}{x}{}{{y}}^{{2}}{+}{{y}}^{{2}}{+}{4}{}{x}{-}{4}⟩$ (5)

In this example, we use EliminationIdeal to derive trigonometric identities algebraically, starting from an ideal of known relations. The trigonometric functions are enclosed in backquotes to prevent Maple from recognizing them.

 > $\mathrm{TRIG}≔⟨{\mathrm{sin\left(x\right)}}^{2}+{\mathrm{cos\left(x\right)}}^{2}-1,\mathrm{cos\left(x\right)}\mathrm{tan\left(x\right)}-\mathrm{sin\left(x\right)},\mathrm{sin\left(2x\right)}-2\mathrm{sin\left(x\right)}\mathrm{cos\left(x\right)},\mathrm{cos\left(2x\right)}-{\mathrm{cos\left(x\right)}}^{2}+{\mathrm{sin\left(x\right)}}^{2},\mathrm{cos\left(2x\right)}\mathrm{tan\left(2x\right)}-\mathrm{sin\left(2x\right)}⟩$
 ${\mathrm{TRIG}}{≔}⟨{\mathrm{cos\left(2x\right)}}{}{\mathrm{tan\left(2x\right)}}{-}{\mathrm{sin\left(2x\right)}}{,}{-}{2}{}{\mathrm{sin\left(x\right)}}{}{\mathrm{cos\left(x\right)}}{+}{\mathrm{sin\left(2x\right)}}{,}{\mathrm{cos\left(x\right)}}{}{\mathrm{tan\left(x\right)}}{-}{\mathrm{sin\left(x\right)}}{,}{{\mathrm{cos\left(x\right)}}}^{{2}}{+}{{\mathrm{sin\left(x\right)}}}^{{2}}{-}{1}{,}{-}{{\mathrm{cos\left(x\right)}}}^{{2}}{+}{{\mathrm{sin\left(x\right)}}}^{{2}}{+}{\mathrm{cos\left(2x\right)}}⟩$ (6)
 > $S≔\mathrm{EliminationIdeal}\left(\mathrm{TRIG},\left\{\mathrm{tan\left(2x\right)},\mathrm{tan\left(x\right)}\right\}\right)$
 ${S}{≔}⟨{\mathrm{tan\left(2x\right)}}{}{{\mathrm{tan\left(x\right)}}}^{{2}}{-}{\mathrm{tan\left(2x\right)}}{+}{2}{}{\mathrm{tan\left(x\right)}}⟩$ (7)
 > $\mathrm{isolate}\left(\mathrm{op}\left(\mathrm{Generators}\left(S\right)\right),\mathrm{tan\left(2x\right)}\right)$
 ${\mathrm{tan\left(2x\right)}}{=}{-}\frac{{2}{}{\mathrm{tan\left(x\right)}}}{{{\mathrm{tan\left(x\right)}}}^{{2}}{-}{1}}$ (8)
 > $T≔\mathrm{EliminationIdeal}\left(\mathrm{TRIG},\left\{\mathrm{cos\left(2x\right)},\mathrm{tan\left(x\right)}\right\}\right)$
 ${T}{≔}⟨{\mathrm{cos\left(2x\right)}}{}{{\mathrm{tan\left(x\right)}}}^{{2}}{+}{{\mathrm{tan\left(x\right)}}}^{{2}}{+}{\mathrm{cos\left(2x\right)}}{-}{1}⟩$ (9)
 > $\mathrm{isolate}\left(\mathrm{op}\left(\mathrm{Generators}\left(T\right)\right),\mathrm{cos\left(2x\right)}\right)$
 ${\mathrm{cos\left(2x\right)}}{=}\frac{{-}{{\mathrm{tan\left(x\right)}}}^{{2}}{+}{1}}{{{\mathrm{tan\left(x\right)}}}^{{2}}{+}{1}}$ (10)

 See Also