MA341D Homework assignment 1
Due on February 4, 2016
1. (a) Rewrite the following polynomial in the decreasing order of terms using the
standard glex order (with x1 < x2 < x3 ): 2x1 x2 + 3x2 x1 + x1 x3 + x32 − x21 x23 + x33 ;
(b) Answer the same question for the glex order with x3 < x2 < x1 .
(c) Answer the same question for the order glex(4,2,1) defined using the weights
wt(x1 ) = 4, wt(x2 ) = 2, wt(x3 ) = 1.
2. (a) For two noncommutative monomials m, m0 in x1 , x2 , we define m ≺ m0 if
• m has fewer occurrences of x2 than m0 , or
• m has the same number of occurrences of x2 as m0 , and m0 = mm00 for some
m00 6= 1, or
• m has the same number of occurrences of x2 as m0 , and in the first position
where the words m and m0 differ, the monomial m has the letter x2 (and the
monomial m0 has the letter x1 ).
Prove that ≺ is a well-order, and that it is admissible.
(b) Does there exist an admissible order of noncommutative monomials in x1 , x2 , x3
for which
x1 x2 > x23 , x2 x3 > x21 , x3 x1 > x22 ?
3. (a) Consider the noncommutative monomial x2 x1 x2 x1 . This polynomial has two
different divisors x2 x1 equal to the leading monomial of x1 x2 − x2 x1 (for the standard
glex order), so there are at least two different ways the long division algorithm may
proceed. Describe all possible ways for the long division to proceed. Do they lead to
the same or different results?
(b) Answer the same question for the long division of x32 by the noncommutative
polynomial x21 − x1 x2 + x22 .
4. (a) Prove that the quotient algebra F hx1 , x2 , x3 i/(x1 − x2 , x1 − x3 ) is isomorphic to
the algebra F [x].
(b) Show that the noncommutative polynomials x1 −x2 , x1 −x3 form a Gröbner basis
of the ideal they generate for the glex order with x1 < x2 < x3 , but do not form a
Gröbner basis of the ideal they generate for the glex order with x1 > x2 > x3 .