Besides the tdeg monomial ordering for "fast" normal form computation and the plex monomial ordering for "slow" triangularization, lexdeg orderings are available for more efficient elimination. For the purpose of generality, the Groebner package implements weighted monomial orderings (wdeg) and matrix-defined monomial orderings. Products of orders and user-defined orderings are also available.
For our first example, we find the implicit equation of a circle from its rational parametrization by eliminating the parameter .
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First we convert to polynomials and clear the fractions.
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As a second example, here is a calculation that is performed far faster by using lexdeg than by using plex. Assume that we want to find the extrema of
on the sphere
Using the method of Lagrange multipliers, we compute
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and we eliminate the multiplier by the use of an appropriate monomial order
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In the general case, we would have to compute numerical estimates and substitute into . Here, we can go further and compute all the possible values for by the elimination of and .
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Substituting into the system yields the possible extremal points.
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Finally, we get the minimum and maximum values.
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Similar calculations using plex require a significantly longer amount of time.