eliminate variables from an ideal (subring intersection) - 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 kX.

• 

Note: You cannot use the Intersect command to compute this result.  For any variables X, the polynomial ring kX 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

withPolynomialIdeals&colon;

J:=x2y&comma;y2&plus;1

J:=y2&plus;1&comma;x2y

(1)

EliminationIdealJ&comma;x

x4&plus;1

(2)

EliminationIdealJ&comma;y

y2&plus;1

(3)

K:=x1&plus;t21t2&comma;y1&plus;y22t

K:=yy2&plus;12t&comma;xt2&plus;1&plus;t21

(4)

EliminationIdealK&comma;x&comma;y

xy6&plus;y6&plus;2xy4&plus;2y4&plus;xy2&plus;y2&plus;4x4

(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.

TRIG:=`sin(x)`2&plus;`cos(x)`21&comma;`cos(x)``tan(x)``sin(x)`&comma;`sin(2x)`2`sin(x)``cos(x)`&comma;`cos(2x)``cos(x)`2&plus;`sin(x)`2&comma;`cos(2x)``tan(2x)``sin(2x)`

TRIG:=cos(2x)tan(2x)sin(2x)&comma;2cos(x)sin(x)&plus;sin(2x)&comma;cos(x)tan(x)sin(x)&comma;cos(x)2&plus;sin(x)21&comma;cos(x)2&plus;sin(x)2&plus;cos(2x)

(6)

S:=EliminationIdealTRIG&comma;`tan(x)`&comma;`tan(2x)`

S:=tan(2x)tan(x)2tan(2x)&plus;2tan(x)

(7)

isolateopGeneratorsS&comma;`tan(2x)`

tan(2x)&equals;2tan(x)tan(x)21

(8)

T:=EliminationIdealTRIG&comma;`cos(2x)`&comma;`tan(x)`

T:=cos(2x)tan(x)2&plus;tan(x)2&plus;cos(2x)1

(9)

isolateopGeneratorsT&comma;`cos(2x)`

cos(2x)&equals;tan(x)2&plus;1tan(x)2&plus;1

(10)

See Also

Groebner[Basis], isolate, PolynomialIdeals, PolynomialIdeals[Generators], PolynomialIdeals[Intersect]


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