Spring 2016, problem 12

Let $g(z) = \frac{a_1z + b_1}{c_1z + d_1}$ with $a_1,b_1,c_1,d_1 \in \mathbf{Z}$, $(a_1-d_1)^2 + 4b_1c_1 \ne 0$, and at least one of $c_1,d_1$ is non-zero. Now suppose that $g^n(z) = z$ and that $g^j(z) \ne z$ for $0<~$$j<n$ and define $\textrm{sgn}(g(z)) = n$. Find all possible values of $\textrm{sgn}(g(z))$.


5 years ago

A partial solution with $\Delta= (a_1-d_1)^2+4b_1c_1>0$; write $g^n(z)=\frac {a_nz+b_n}{c_nz+d_n}$ $ M_n=\begin{pmatrix} a_n & c_n \\ b_n & d_n \end{pmatrix}, \; M=M_1$; then an easy induction shows $M_n=M^n;$ now $det(M-xI)$ has discriminant $\Delta>0$ hence $M$ has eigenvalues $\alpha,\beta \in\mathbb{R}$; so $\exists P, \; M^n=P^{-1} \begin{pmatrix} \alpha^n & 0 \\ 0 & \beta^n \end{pmatrix}P=I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\Rightarrow~\alpha, \beta \in\big\{-1, 1\big\}\Rightarrow$ $n$ is $1$ or $2$, which works for $g(z)=z$ or $-z$.

There is a slight wrinkle here, namely that more than just the identity matrix represents the identity transformation, so you need to deal with the possibility that $M^n=7I$. i suspect (but cannot yet prove) that because the eigenvalues are quadratic irrationalities, there wont be any cases where they arent roots of unity (in which case $n=1,2,3$, with 4th and 6th roots of unity not contributing new cases because lower powers yield $-I$. Howecer, I do not yet have more than my intuition to go on right now.

Anonymouse 4 years ago
5 years ago

Is $z$ the variable here, or are $a_1, b_1, c_1, d_1$ all variables as we look for such functions, too? What is the domain of $g$? Does the minimum period have to work for all $z$? or just one particular $z$?

Yea this is sloppy authored math problems. Who the hell is comeing up with these ?! Certainly not the MAA .the purdue pow used to be good when a ' panel ' would submit many possible gems (problems)and then all decide on one for the ' audience '. Lately its been * chunks of coal * lol --> we want better ones ;)

Samurai 5 years ago

Agreed. Back when I was an undergrad here, the quality of the problems were better, as was the nature of posting the solutions. It honestly wouldn't be difficult for them to post the problems from before then and recycle them as they go.

ibachniv 5 years ago

Yep ' recycling ' is great as theres so many math problems out there one cant remember how to solve them by memory !

Samurai 5 years ago

Here is one of my favorite ppow :At time 0 each of the positions 1, 2,...,n on the real line is occupied by a robot, and position 0 is occupied by the prey. At time k (k = 1,...,n) one of the robots, selected at random, jumps one unit to the left, unless that robot has been previously disabled, in which case nothing happens. If it lands on position 0, the prey is destroyed; but if it lands on another robot, both robots are disabled. Assuming that each robot is selected to jump exactly once and that all n! jumping orders are equally likely, find the probability pn, that the prey is eventually destroyed and also find lim n→∞pn. (Your answer for pn need not be in closed form.)

Samurai 5 years ago