Math 662
Homework 1
Spring 2014
Drew Armstrong
Problem 0 (Drawing Pictures). Sketch the curves y 2 = f (x) in R2 for the following
polynomials f (x) R[x]: f (x) = x3 and f (x) = (x + 1)(x2 + ) for < 0, = 0, > 0. [Hint:
First sketch y = f (x) then sketch y
Math 662
Homework 3
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) What does the curve y 2 = x2 (x+1) look like at innity?
We dene n-dimensional real projective space RP n as the set of lines through the origin in Rn+1 .
That is, we dene
RP n :=
Math 662
Homework 2
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) The equation y 2 = x3 x denes a curve in the
complex plane C2 . What does it look like? Unfortunately we can only see real things, so
we substitute x = a + ib and y = c + id with
Math 662
Final Exam
Spring 2014
Drew Armstrong
1. (Galois Connections) Let R be any ring. Given any set of points S K n we dene
a set of polynomials I(S) := cfw_f R[x1 , . . . , xn ] : f () = 0 for all S, and given any
set of polynomials T R[x1 , . . . ,
Math 662
Homework 4
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) We have drawn algebraic curves in R2 , C2 and RP 2 . Now
we will try to draw an algebraic curve in CP 2 . Let , , C be distinct complex numbers
and consider the polynomial
f (x,
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Math 662
Homework 4
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) We have drawn algebraic curves in R2 , C2 and RP 2 . Now
we will try to draw an algebraic curve in CP 2 . Let , , C be distinct complex numbers
and consider the polynomial
f (x,
Math 662
Homework 2
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) The equation y 2 = x3 x denes a curve in the
complex plane C2 . What does it look like? Unfortunately we can only see real things, so
we substitute x = a + ib and y = c + id with
Math 662
Homework 3
Spring 2014
Drew Armstrong
Problem 0. (Drawing Pictures) What does the curve y 2 = x2 (x+1) look like at innity?
We dene n-dimensional real projective space RP n as the set of lines through the origin in Rn+1 .
That is, we dene
RP n :=
Math 662
Homework 1
Spring 2014
Drew Armstrong
Problem 0 (Drawing Pictures). Sketch the curves y 2 = f (x) in R2 for the following
polynomials f (x) R[x]: f (x) = x3 and f (x) = (x + 1)(x2 + ) for < 0, = 0, > 0. [Hint:
First sketch y = f (x) then sketch y