Answer by Douglas Lind for Monic polynomial with integer coefficients with...
This is an adelic comment based on algebraic dynamics. Each such polynomial $f(x)\in\mathbb{Z}[x]$ induces an algebraic dynamical system (i.e. an automorphism of a compact abelian group). If $f(x)$ has...
View ArticleAnswer by Nikita Sidorov for Monic polynomial with integer coefficients with...
Just a couple of minor top-ups to Dmitri's nice answer. For each even $n\ge2$ the polynomial $p_n(x)=x^n-x^{n-1}-\dots-x+1$ is a Salem polynomial.It is not known whether for any $\delta>0$ there...
View ArticleAnswer by Benjamin Dickman for Monic polynomial with integer coefficients...
First, you are right: if a monic polynomial's roots are all on the unit circle, then every root is a root of unity. This follows from a result of Kronecker's (see: Zwei Sätze über Gleichungen mit...
View ArticleAnswer by Timothy Foo for Monic polynomial with integer coefficients with...
For a class of concrete examples with at least asymptotically more than $n/2$ zeros on the unit circle, the Fekete polynomials, which were just mentioned recently by Franz Lemmermeyer at this question...
View ArticleAnswer by Dmitri Panov for Monic polynomial with integer coefficients with...
There exist irreducible monic polynomials such that all their roots apart from two lie on the unique circle (and are not roots of unity). Such polynomials can be chosen among Salem polynomials and they...
View ArticleMonic polynomial with integer coefficients with roots on unit circle, not...
There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.Now, for a monic polynomial of degree $n$, this...
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