are ideals in | > |
MATH 392 -- Seminar in Computational Commutative Algebra
November 1, 2006
Gr"obner basis method for ideal intersections.
Recall that if are ideals in
then we can compute the intersection of I and J
by computing
This can be done by a Gr"obner basis computation with
respect to any monomial order that has the elimination
property with respect to t (any monomial containing
t is greater than all monomials not containing t).
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| > | ![IdI := [x^2*y-z, x*y-z+1, x^2+y^3-1]; 1](images/IdInt_5.gif){kind=small} |
![[x^2*y-z, x*y-z+1, x^2+y^3-1]](images/IdInt_6.gif){kind=small}
| > | ![IdJ := [y^2-x+2, x+y+z]; 1](images/IdInt_7.gif){kind=small} |
![[y^2-x+2, x+y+z]](images/IdInt_8.gif){kind=small}
| > | ![EI := [seq(t*IdI[i], i = 1 .. nops(IdI)), seq((1-t)*IdJ[j], j = 1 .. nops(IdJ))]; 1](images/IdInt_9.gif){kind=small} |
![[t*(x^2*y-z), t*(x*y-z+1), t*(x^2+y^3-1), (1-t)*(y^2-x+2), (1-t)*(x+y+z)]](images/IdInt_10.gif){kind=small}
![[t*(x^2*y-z), t*(x*y-z+1), t*(x^2+y^3-1), (1-t)*(y^2-x+2), (1-t)*(x+y+z)]](images/IdInt_11.gif){kind=small}
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| > | ![for i to nops(B) do has(B[i], t) end do; 1](images/IdInt_35.gif){kind=small} |
The first three polynomials in B form a Gr"obner basis for 
(with respect to lex, x > y > z,
by the Elimination Theorem from Chapter 3).
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