MA3417 Homework assignment 1
Due on October 10, 2013
Unless otherwise specified, we assume in questions below that the variables are ordered
with x > y > z, respectively x1 > x2 > · · · > xn .
1. For each of the following polynomials, rewrite it in the decreasing order of terms
using the LEX order, the DEGLEX order, and the DEGREVLEX order:
(a) 2x + 3y + z + x2 − z 2 + x3 ;
(b) 2x2 y 8 − 3x5 yz 4 + xyz 3 − xy 4 .
2. Do the the previous question for the order of variables z > y > x.
3. Each of the following polynomials is written in the decreasing order of terms using
(exactly) one of the LEX order, the DEGLEX order, and the DEGREVLEX order.
Determine which order was used in each case:
(a) 7x2 y 4 z − 2xy 6 + x2 y 2 ;
(b) xy 3 z + xy 2 z 2 + x2 z 3 ;
(c) x4 y 5 z + 2x3 y 2 z − 4xy 2 z 4
4. Let A = (aij ) be P
a matrix in its reduced row echelon form. Let us consider linear
polynomials fi = j aij xj . Find an order for which fi form a Gröbner basis. (Hint:
choose an ordering of variables wisely, and use the LEX order for that chosen ordering
of variables).
5. Show that for the ideal I = (y − x2 , z − x3 ) in C[x, y, z], the polynomials y − x2 , z − x3
that generate it do not form a Gröbner basis for the LEX order with x > y > z.
6. Show that for the ideal I from the previous question, the polynomials y − x2 , z − x3
that generate it form a Gröbner basis for the LEX order with z > y > x.
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