\documentclass[12pt]{amsart}

\newcommand{\K}{\bf K}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\A}{\mathbb{A}}

\newcommand{\Z}{\mathcal{Z}}
\newcommand{\I}{\mathcal{I}}

\newtheorem*{lemma}{Lemma}

\begin{document}

\title{Math 172 (Su):  HW number 10}
\maketitle

As usual, ``R'' means read, but do not do the problem (do it in your head).

\vfill

1. Let $\K$ be an infinite field, and $f$ and $g$ in
   $\K[x_1,...,x_n]$.  Show that $f=g$ as polynomials if and only if
   $f,g:\A \rightarrow K$ are the same function.

\vfill

2. If $W = \Z(f_1,...,f_s)$ and $V=\Z(g_1,...,g_t)$ 
   are algebraic sets in $A^n$, show that $W\cup V$ and
   $W\cap V$ are algebraic sets.

\vfill

R3. Show that every finite subset of $\A^n$ is an algebraic set.

\vfill

4. Show that $X=\{ (x,x) : x \in \R, x\neq 1 \}$ is {\em not} an
   algebraic set in $\R^2$.  Hint: if $f \in \R[x,y]$ vanishes on $X$,
   what can be said about $f(1,1)$?

\vfill

5. Let $V$ be an algebraic set in $\A^n$ (over a field $K$).  
   Let $\I(V)$ be the ideal of all polynomials in $\K[x_1,...,x_n]$
   that vanish (evaluate to $0$) on the set $V$.

Call an ideal $I$ in a commutative ring a {\em radical ideal} if
for all $r\in R$, $r^m \in I$ implies $r \in I$.

   Show that $\I(V)$ is a radical ideal.



\vfill

6. For any nonzero polynomials $f$ and $g$, show that:

(a)  $LT(fg)=LT(f)LT(g)$ and $\partial(fg)=\partial(f)+\partial(g)$.

(b)  $\partial(f+g) \leq \max \{ \partial(f), \partial(g)\}$ with equality if
     $\partial(f)\neq \partial(g)$ (where $\leq$ is a monomial order).

\vfill

\end{document}
