\input /profs/cs1/kaltofen/text/def.tex
\twelvepoint
\hbox to \hsize{\hfill\vtop{\halign {\hfill {\bf #}\hfill\cr
65-4962-01 \& 66-4961-01 Computational Abstract Algebra\cr
Homework~6\cr
Due 3/29/1990\cr}}\hfill}
\def\Problem#1{\medskip\noindent{\bf Problem~#1:}}
\Problem1
Apply Sturm's Theorem to determine the number of real roots of $x^4 +
12x^2 + 5x - 9.$
\Problem2
Let $f(x) = x^3 + px + q, p \neq 0.$ Show that $$
f_0 = f, f_1 = 3x^2 + p, f_2 = -2px - 3q, f_3 = -4p^3 - 27q^2$$ is a
Sturm sequence for $f(x)$ for any $[a,b]$ with $f(a)f(b) \neq 0.$ Note
that $f_3 = d$, the disciminant of $f(x)$. Use Sturm's theorem
to prove that $f$ has a single real root or three distinct real root
according as $d < 0$ or $d > 0.$
\Problem3
Define the sequence of {\sl Legendre polynomials} $P_0, P_1, \ldots,
P_n,\ldots $ by the recursive formula $$
nP_n(x) - (2n - 1)xP_{n-1} + (n-1)P_{n-2}(x) = 0$$ where $P_0(x) = 1,
P_1(x) = x.$ Show that $\{P_m,P_{m-1}, \ldots, P_0\}$ is a Sturm sequence
for $P_m$ for the interval $[-1,1].$ Show that $P_m$ has $m$ distinct
real roots in $(-1,1).$
\bye