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MATH 392 -- Seminar in Computational Commutative Algebra 

Generalized Resultants for Elimination 

October 25, 2006 

When we have an ideal generated by several polynomials, 

the generalized resultant is computed by introducing  

new u-variables as follows. 

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We can expand this polynomial, and collect like terms in the u-variables: 

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We claim that each of the coefficients of the monomials  

here is an element of the elimination ideal To verify that 

claim, we will compute a Grobner basis of  I = <FList>: 

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Then "pick off" the coefficients of each monomial in the  u-variables. 

Note that each contains only  y,z 

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Then we divide the coefficients by the Grobner basis: 

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This shows that every one of the coefficients of the generalized resultant is 

in the elimination ideal.    

CAUTION:  While these polynomials are in the elimination ideal, they do  

not necessarily generate the elimination ideal.  For instance, in this case,  

we can see that the coefficients generate an ideal strictly contained in the  

elimination ideal: 

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Recall, reduced Grobner bases are unique.  These polynomials are not the 

same as the Grobner basis for the elimination ideal obtained from B above. 

For instance, the univariate polynomial in z  in BB is a multiple of the  

univariate polynomial in z from B: 

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