View Single Post
2017-07-08, 13:04   #4
Dr Sardonicus

Feb 2017
Nowhere

23·739 Posts

Quote:
Originally Posted by Nick
Originally Posted by Dr Sardonicus
Quote:
 ...Clearly the lead coefficient is 1...
I'm not sure I follow you there...
[After a decent amount of sleep]

Fascinating, how I messed that up. I didn't even mean to post that last response. I must have hit "Submit reply" by mistake...

What is clear about a reciprocal polynomial, is that the lead coefficient is equal to the constant term. [The coefficients of any two terms of complementary degree are always equal.]

Beyond that... well, the zero polynomial is a reciprocal polynomial, and there's no way to make its lead coefficient equal to 1. So, you have to assume F(z) is not the zero polynomial. Given that, you can, of course, always assume it's monic, since it will have a non-zero lead coefficient, and you can just divide by it. And then, F(z) has lead coefficient and constant term both equal to 1.

But that's not really what I had in mind. I'm only interested here in monic polynomials with integer coefficients. And there's no way around having to assume that F(z) is monic with integer coefficients -- after all, e.g. z^2 - (5/2)*z + 1 is a reciprocal polynomial.

Last fiddled with by Dr Sardonicus on 2017-07-08 at 13:07