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Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
Silberstein: Prime Polynomials From: Keith Conrad To: William Stein Date: 2004-07-25 01:04 am William, Hi. I stumbled onto Aaron Silberstein's description of his Meccah computation on the computation log. He was doing computations connected with work by me, Brian, and Rob Gross. (I had asked him to check some numerics for us, and after a bit he wrote to you.) The write-up by Aaron, which looks like it may be an email, has a number of mistakes, e.g., he writes g(T) when he meant g(X), and he did not credit Rob Gross for anything. For that and other reasons, I'd prefer that his summary of the computations connected to our work not be posted on the computation log. Could you just give a title for the computations and leave it at that, as with Samit and others? Keith

Aaron Silberstein

MECCAH Computation

I am computing a few things. They all are calculations in a function
field over a finite field. So let's fix a prime power q = p^k. Then
consider g in F_q(T)[X] (that is, g(T) is a polynomial in F_q(T)[X]).
What can you say about the number of prime polynomials of the form
g(f(X)), where f is in F_q[X] of degree n and where g is fixed? Keith
Conrad and his brother Brian have some asymptotic results, and they
want to verify them. Some interesting patterns occur; for example, in
some degrees, there are no such prime polynomials.

There is another calculation I am running, which is turning up some
interesting pattern. Let mu be the Mobius mu function over F_q[X]
(that is, it is -1 to the number of prime polynomials dividing the
argument, and 0 if it is not squarefree). Then what is \sum_{\deg f =
n} mu(g(f(x)))? We are finding some surprising patterns in many cases,
especially when p = 2 and g = T^2 + X; in this case, for example, when
k = 1, 2, 3, for n = 2 mod 3, this sum is always zero. No one can
account for this so far. So I'm running calculations so that we can
verify if this is just some anomaly, and to see what asymptotics might
be for these calculations. I called you because I know you have
computing power, and calculations right now are obnoxiously slow on the
servers on which I can run MAGMA.

I thank you again for your help, and if you have any more questions
feel free to email me ([email protected]) or Keith
([email protected]).

I am computing a few things. They all are calculations in a function
field over a finite field. So let's fix a prime power q = p^k. Then
consider g in F_q(T)[X] (that is, g(T) is a polynomial in F_q(T)[X]).
What can you say about the number of prime polynomials of the form
g(f(X)), where f is in F_q[X] of degree n and where g is fixed? Keith
Conrad and his brother Brian have some asymptotic results, and they
want to verify them. Some interesting patterns occur; for example, in
some degrees, there are no such prime polynomials.

There is another calculation I am running, which is turning up some
interesting pattern. Let mu be the Mobius mu function over F_q[X]
(that is, it is -1 to the number of prime polynomials dividing the
argument, and 0 if it is not squarefree). Then what is \sum_{\deg f =
n} mu(g(f(x)))? We are finding some surprising patterns in many cases,
especially when p = 2 and g = T^2 + X; in this case, for example, when
k = 1, 2, 3, for n = 2 mod 3, this sum is always zero. No one can
account for this so far. So I'm running calculations so that we can
verify if this is just some anomaly, and to see what asymptotics might
be for these calculations. I called you because I know you have
computing power, and calculations right now are obnoxiously slow on the
servers on which I can run MAGMA.

I thank you again for your help, and if you have any more questions
feel free to email me ([email protected]) or Keith
([email protected]).