Problem Set 1 Solutions
Math 480/550
(1) Using Denition 2 from section 1.1 in the book, dene polynomial multiplication, addition,
and subtraction, and show that k [x1 , , xn ] is a ring.
Solution: Write a polynomial f k [x1 , xn ] as f =
a x , for Zn 0 an
Problem Set 2 Solutions
(10 pts)
Math 480/550
(1) The consistency problem in C. In this problem, let k = C, and consider a collection
of polynomials f1 , . . . , fr k [x]. The collection is inconsistent if the polynomials have
no common solutions: V(f1 ,
Problem Set 3
Due 6 March, 2009
Math 480/550
Solutions are to be submitted in PDF form via email to gun@math.upenn.edu by 5pm on
6 March, 2009.
(10 pts)
(1) Let
denote lex order on Nn . Is it always true that for any two monomials x
x ,
there are a nite n
Problem Set 4
Due 27 March, 2009
Math 480/550
Solutions are to be submitted in PDF form via email to gun@math.upenn.edu by 5pm.
(10 pts)
(1) Consider the system in Figure 1. It consists of three rigid rods (in red, with lengths 2, 2,
and 4 as shown) conne
Problem Set 5
Due 10 April, 2009
Math 480/550
Solutions are to be submitted in PDF form via email to gun@math.upenn.edu by 5pm.
(10 pts)
(1) The algebraic construction of derivatives. Let k be an arbitrary eld, and dene the
ring of dual numbers as R = k [
Problem Set 5
Due 4 May, 2009
Math 480/550
Solutions are to be submitted in PDF form via email to gun@math.upenn.edu by 5pm.
In the problems below, Q k and the k is ommitted from Pn . We only consider the Zariski
k
topology on varieties.
(60 pts)
(1) Let