LECTURE 1. SYSTEMS OF ALGEBRAIC EQUATIONS
1. For which elds k do the systems
S = cfw_ i (T1 , . . . , Tn ) = 0i=1,.,n , and S = cfw_
Tji = 0i=1,.,n
dene the same a ne algebraic varieties? Here i (T1 , . . . , Tn ) denotes the eleme
Proof. Let us prove the rst part. If V is irreducible, then the assertion is obvious.
Otherwise, V = V1 [ V2 , where Vi are proper closed subsets of V . If both of
them are irreducible, the assertion is true. Otherwise, one of them, say V1 is
Physical Science Final Exam
Type your answers in the blank for each question. Do NOT use the internet - use ONLY the textbook to
create solutions in your own words.
1. A car that increases its speed from 20 km/h to 100 km/h undergoes
CMN 251: Organizational Communication
Final Case Development & Analysis Paper
For this assignment, you are required to create and analyze an original case.
Working with 1-2 of your peers, you will need to develop a fictional organizational
Recently, the Genesis Energy is going to make a contract with Sensible Energy to undertake the
project of operating expansion. But before going along with this project, study of projects
feasibility is needed and therefore estimation has
HOMEWORK ASSIGNMENT #6
1. Evaluate the following SCILAB expressions yourself (Dont use SCILAB to
help! This is working on the concept of precedence). You can do this all in
SCILAB, but for this problem use comments '/' to prevent execution. When
You work for a HR consulting company and an organization (the same company you
have been writing about during this course) has hired your firm to conduct an HRM
analysis and make recommendations to better align HR practices to the key business
For the Unit VIII assignment, please refer to Section 5.4 of the text.
Monica works at a regional weather office on the Atlantic coast. She notes (from the office records) that
hurricanes have made landfall on the coast somewhere near their city of Johnst
Global Warming In April 2011, a Yale/George Mason poll of 1010 US adults found that 40% of the
people responding believed that scientists disagreed about whether global warming exists. They
reported a 95% confidence interval of 37
Proof. Let f : X ! Y be a morphism. It denes a homomorphism
fO(X ) : Homk (O(X ), O(X ) ! Homk (O(Y ), O(X ).
The image of the identity homomorphism idO(X ) is a homomorphism : O(Y ) !
O(X ). Let us show that ( ) = f . Let 2 X (K ) = Homk (O(X ), K ).
algebras: O(Y ) = k [T1 ] ! O(X ). Every such homomorphism is determined by
its value at T1 , i.e. by an element of O(X ). This gives us one more interpretation
of the elements of the coordinate algebra O(X ). This time they are morphisms
from X to A1
k as a subeld. Note that A is nitely generated as a k -algebra (because k [T ]
is). Suppose we show that A is an algebraic extension of k . Then we will be
able to extend the inclusion k K to a homomorphism A ! K (since K is
algebraically closed), the c
1, . . . , m(r), be the polynomials from I which have the highest coe cient equal
to ai,r . Next, we consider the union J of the ideals Jr . By multiplying a
polynomial F by a power of Tn we see that Jr Jr+1 . This immediately implies
that the union J i
Example 1.1. 1. The system S = cfw_0 k [T1 , . . . , Tn ] denes an a ne algebraic variety denoted by An . It is called the a ne n-space over k . We have, for
any k -algebra K ,
Sol(cfw_0; K ) = K n .
2. The system 1 = 0 denes the empty a ne algebraic
Systems of algebraic equations
The main objects of study in algebraic geometry are systems of algebraic equations and their sets of solutions. Let k be a eld and k [T1 , . . . , Tn ] = k [T ] be
the algebra of polynomials in n variables over k .
Next we shall show that the set of algebraic k -subsets in K n can be used
to dene a unique topology in K n for which these sets are closed subsets. This
follows from the following:
Proposition 2.7. (i) The intersection \s2S Vs of any family cfw_Vs s2S
Morphisms of a ne algebraic
In Lecture 1 we dened two systems of algebraic equations to be equivalent if
they have the same sets of solutions. This is very familiar from the theory of
linear equations. However this notion is too strong
Irreducible algebraic sets and
We know that two a ne algebraic k -sets V and V 0 are isomorphic if and only if
their coordinate algebras O(V ) and O(V 0 ) are isomorphic. Assume that both of
these algebras are integral domains
X ! Y of algebraic varieties denes a map fK : X (K ) = V ! Y (K ) =
W of the algebraic sets. So it is natural to take for the denition of regular
maps of algebraic sets the maps arising in this way. We know that f is given
by a homomorphism of k -algeb
There are many issues revolving around investor protection in the world. To start with countries with
common law tradition such as Canada, UK and US have highly protective investors policies compared to
civil law countries such as Germany and France. Howe