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.

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\&colon;$

$J≔⟨x^2-y\&comma;y^2+1⟩$

$J≔⟨y^2+1\&comma;x^2-y⟩$

$\mathrm{EliminationIdeal}\left(J\&comma;\left\{x\right\}\right)$

$⟨x^4+1⟩$

$\mathrm{EliminationIdeal}\left(J\&comma;\left\{y\right\}\right)$

$⟨y^2+1⟩$

$K≔⟨x\left(1+t^2\right)-\left(1-t^2\right)\&comma;y\left(1+y^2\right)-2t⟩$

$K≔⟨y\left(y^2+1\right)-2t\&comma;x\left(t^2+1\right)+t^2-1⟩$

$\mathrm{EliminationIdeal}\left(K\&comma;\left\{x\&comma;y\right\}\right)$

$⟨x{y}^6+y^6+2x{y}^4+2y^4+x{y}^2+y^2+4x-4⟩$

$\mathrm{TRIG}≔⟨{\mathrm{`sin(x)`}}^2+{\mathrm{`cos(x)`}}^2-1\&comma;\mathrm{`cos(x)`}\mathrm{`tan(x)`}-\mathrm{`sin(x)`}\&comma;\mathrm{`sin(2x)`}-2\mathrm{`sin(x)`}\mathrm{`cos(x)`}\&comma;\mathrm{`cos(2x)`}-{\mathrm{`cos(x)`}}^2+{\mathrm{`sin(x)`}}^2\&comma;\mathrm{`cos(2x)`}\mathrm{`tan(2x)`}-\mathrm{`sin(2x)`}⟩$

$\mathrm{TRIG}≔⟨\mathrm{cos(2x)}\mathrm{tan(2x)}-\mathrm{sin(2x)}\&comma;-2\mathrm{sin(x)}\mathrm{cos(x)}+\mathrm{sin(2x)}\&comma;\mathrm{cos(x)}\mathrm{tan(x)}-\mathrm{sin(x)}\&comma;{\mathrm{cos(x)}}^2+{\mathrm{sin(x)}}^2-1\&comma;-{\mathrm{cos(x)}}^2+{\mathrm{sin(x)}}^2+\mathrm{cos(2x)}⟩$

$S≔\mathrm{EliminationIdeal}\left(\mathrm{TRIG}\&comma;\left\{\mathrm{`tan(2x)`}\&comma;\mathrm{`tan(x)`}\right\}\right)$

$S≔⟨\mathrm{tan(2x)}{\mathrm{tan(x)}}^2-\mathrm{tan(2x)}+2\mathrm{tan(x)}⟩$

$\mathrm{isolate}\left(\mathrm{op}\left(\mathrm{Generators}\left(S\right)\right)\&comma;\mathrm{`tan(2x)`}\right)$

$\mathrm{tan(2x)}=-\frac{2\mathrm{tan(x)}}{{\mathrm{tan(x)}}^2-1}$

$T≔\mathrm{EliminationIdeal}\left(\mathrm{TRIG}\&comma;\left\{\mathrm{`cos(2x)`}\&comma;\mathrm{`tan(x)`}\right\}\right)$

$T≔⟨{\mathrm{tan(x)}}^2\mathrm{cos(2x)}+{\mathrm{tan(x)}}^2+\mathrm{cos(2x)}-1⟩$

$\mathrm{isolate}\left(\mathrm{op}\left(\mathrm{Generators}\left(T\right)\right)\&comma;\mathrm{`cos(2x)`}\right)$

$\mathrm{cos(2x)}=\frac{-{\mathrm{tan(x)}}^2+1}{{\mathrm{tan(x)}}^2+1}$