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A.3.3 Elimination

Elimination is the algebraic counterpart of the geometric concept of projection. If is a polynomial map, the Zariski-closure of the image is the zero-set of the ideal where that is, of the ideal j obtained from J by eliminating the variables This can be done by computing a Groebner basis for J with respect to a (global) product ordering where the block of t-variables precedes the block of x-variables, and then selecting those polynomials which do not contain any t. Alternatively, we may use a global monomial ordering with extra weight vector (see Extra weight vector), assigning to the t-variables a positive weight and to the x-variables weight 0.

Since elimination is expensive, it may be useful to use a Hilbert-driven approach to the elimination problem (see Groebner basis conversion):

First compute the Hilbert function of the ideal w.r.t. a fast ordering (e.g., dp), then make use of it to speed up the computation by a Hilbert-driven elimination which uses the intvec provided as third argument.

In SINGULAR the most convenient way is to use the eliminate command. In contrast to the first method, with eliminate the result needs not be a standard basis in the given ordering. Hence, there may be cases where the first method is the preferred one.

WARNING: In the case of a local or a mixed ordering, elimination needs special care. f may be considered as a map of germs but even if this map germ is finite, we are in general not able to compute the image germ because for this we would need an implementation of the Weierstrass preparation theorem. What we can compute, and what eliminate actually does, is the following: let V(J) be the zero-set of J in then the closure of the image of V(J) under the projection (Note that this germ contains also those components of V(J) which meet the fiber of pr outside the origin.) This is achieved by an ordering with the block of t-variables having a global ordering (and preceding the x-variables) and the x-variables having a local ordering.

In any case, if the input is weighted homogeneous (=quasihomogeneous), the weights given to the variables should be chosen accordingly. SINGULAR offers a function weight which proposes, given an ideal or module, integer weights for the variables, such that the ideal, resp. module, is as homogeneous as possible with respect to these weights. The function finds correct weights, if the input is weighted homogeneous (but is rather slow for many variables). In order to check, whether the input is quasihomogeneous, use the function qhweight, which returns an intvec of correct weights if the input is quasihomogeneous and an intvec of zeros otherwise.

Let us give three examples:

  1. First we compute the equations of the simple space curve 'T[7]' consisting of two tangential cusps given in parametric form.
  2. We compute weights for the equations such that the equations are quasihomogeneous w.r.t. these weights.
  3. Then we compute the tangent developable of the rational normal curve in

 


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