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Math 380, Spring 2018, Assignment 3

2018-02-12T19:21:04Z

Vladimir.Parfenov:

__NOTOC__
''No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....''

: - Morris Kline, ''Mathematics in Western Culture''

==Read:==

# Section 1.4.

==Carefully define the following terms, then give one example and one non-example of each:==

# Ideal (in $\mathsf{k}[x_1,\dots,x_n]$).
# $\left\langle S\right\rangle$ (the ideal generated by a set $S$ of polynomials).
# $\left\langle f_1,\dots,f_s\right\rangle$ (the ideal generated by the finite set $\{f_1,\dots,f_s\}$).
# $\mathbb{I}(T)$ (the ''ideal'' of the set of points $T\subseteq\mathsf{k}^n$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating $\mathbb{V}(S)$ to $\mathbb{V}(\left\langle S\right\rangle)$.
# Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle\subseteq\left\langle g_1,\dots,g_t\right\rangle$.
# Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle=\left\langle g_1,\dots,g_t\right\rangle$.
# Theorem concerning the ''inclusion-reversing'' and ''inflationary'' character of the fundamental pairing $(\mathbb{V},\mathbb{I})$.

==Solve the following problems:==

# Section 4, problems 1, 3, 5, 6(a), 6(b), 7, and 8. ''(In problem 6, the word "basis" means "generating set.")''
# Prove that, for any set $T\subseteq\mathsf{k}^n$, the set $\mathbb{I}(T)$ is always an ''ideal'' of $\mathsf{k}[x_1,\dots,x_n]$.
# Working in $\mathbb{R}^1$, describe $\mathbb{V}(\mathbb{I}(\mathbb{Z}))$. ''(In topological language, this exercise shows that the "Zariski closure" of a set may be quite different from its ordinary closure. This is because the Zariski topology is extremely "coarse.")''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==
This came up because of exercise 1.4.7 but is just a general question: Are we allowed to assume we're working in the smallest affine space that will fit around the polynomials concerned i.e: does the answer boil down to the same thing whether you do your computations in k[x,y] or k[x,y,z,..,w]?

==Solutions:==

Math 380, Spring 2018, Assignment 3

2018-02-12T19:19:46Z

Vladimir.Parfenov:

__NOTOC__
''No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....''

: - Morris Kline, ''Mathematics in Western Culture''

==Read:==

# Section 1.4.

==Carefully define the following terms, then give one example and one non-example of each:==

# Ideal (in $\mathsf{k}[x_1,\dots,x_n]$).
# $\left\langle S\right\rangle$ (the ideal generated by a set $S$ of polynomials).
# $\left\langle f_1,\dots,f_s\right\rangle$ (the ideal generated by the finite set $\{f_1,\dots,f_s\}$).
# $\mathbb{I}(T)$ (the ''ideal'' of the set of points $T\subseteq\mathsf{k}^n$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating $\mathbb{V}(S)$ to $\mathbb{V}(\left\langle S\right\rangle)$.
# Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle\subseteq\left\langle g_1,\dots,g_t\right\rangle$.
# Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle=\left\langle g_1,\dots,g_t\right\rangle$.
# Theorem concerning the ''inclusion-reversing'' and ''inflationary'' character of the fundamental pairing $(\mathbb{V},\mathbb{I})$.

==Solve the following problems:==

# Section 4, problems 1, 3, 5, 6(a), 6(b), 7, and 8. ''(In problem 6, the word "basis" means "generating set.")''
# Prove that, for any set $T\subseteq\mathsf{k}^n$, the set $\mathbb{I}(T)$ is always an ''ideal'' of $\mathsf{k}[x_1,\dots,x_n]$.
# Working in $\mathbb{R}^1$, describe $\mathbb{V}(\mathbb{I}(\mathbb{Z}))$. ''(In topological language, this exercise shows that the "Zariski closure" of a set may be quite different from its ordinary closure. This is because the Zariski topology is extremely "coarse.")''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==
This came up because of exercise 1.4.7 but is just a general question: Are we allowed to assume we're working in the smallest affine space that will fit around the polynomials concerned or does the answer boil down to the same thing whether you do your computations in k[x,y] or k[x,y,z,..,w]?

==Solutions:==

Math 360, Fall 2016, Assignment 1

2016-09-14T19:19:56Z

Vladimir.Parfenov: /* Carefully state the following theorems (you do not need to prove them): */

__NOTOC__
''By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.''

: - Mary Shelley, ''Frankenstein''

==Carefully define the following terms, then give one example and one non-example of each:==

# Cartesian product (of two sets).
# Relation (from $A$ to $B$).
# Function (from $A$ to $B$).
# Domain (of a function).
# Codomain (of a function).
# Image (of a function).
# Injection (a.k.a. ''one-to-one function'').
# Surjection (a.k.a. ''onto function'').
# Bijection.
# Equinumerous (a.k.a. ''equipotent'' or ''having the same cardinality'').

==Carefully state the following theorems (you do not need to prove them):==

# Russell's paradox (this is not exactly a theorem, but it is an important fact).
# Cantor's theorem.

==Solve the following problems:==

# Section 0, problems 1, 5, 7, 11, and 12.
# Prove that the function $f:\mathbb{Z}\rightarrow2\mathbb{Z}$ defined by the formula $f(n) = 2n$ is a bijection. ''(Hint: to prove injectivity, assume that $f(a) = f(b)$ and show that $a=b$. To prove surjectivity, let $b$ be any element of $2\mathbb{Z}$, and find some $a\in\mathbb{Z}$ with $f(a) = b$.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==