A proof of Pisot's dth root conjecture

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a system of infinitely many equations. We present such a system of equations that strictly corresponds with the system for $n$-th of a formal power series of one variable. With help of an example we show that the case of formal power series of m...

Let $p(x) \in C(x)$ be a rational function satisfying the condition $p(0)=1$ and $q$ an integer larger than $1$, in this article we will consider the power expansion of the infinite product $$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$ and study when the sequence $(c_i)_{i \in \mathbf{N}}$ is $q$-automatic. The main result is that for given integers $q \geq 2$ and $d \geq 0$, there exist finitely many polynomials of degree $d$ defined over the field of ...

In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was ...

It is well known that a polynomial $\phi(X)\in \mathbb{Z}[X]$ of given degree $d$ factors into at most $d$ factors in $\mathbb{F}_p$ for any prime $p$. We prove in this paper the existence of infinitely many primes $q$ so that the given polynomial $\phi$(X) splits into exactly $d$ linear factors in $\mathbb{F}_q$ by using only elementary results in field theory and some elementary number theory by proving that $\phi$ splits in $\mathbb{F}_q$ iff $P$ has a root in $\mathbb{F}_...

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over $\mathbb{F}_p$. Let $d\geq 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be irreducible. Further, when the necessary conditions are satisfied, we obtain the probability for $b(x^d)$ to be irreducible.

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies ...

We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new consequences, including a necessary condition for the algebraic solution by radicals of certain irreducible polynomials.

The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. Let $f=x^d+a_1x^{d-1}+\cdots+a_1x \in K[a_1,\ldots,a_{d-1}][x]$ and let $R_i = Res(f,H_i(f))\in K[a_1,\ldots,a_{d-1}]$ be the resultant of $f$ and $H_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R_1,\ldots,R_{d-1}$ are ``in...

Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f in R (where R = 1 + xZ[[x]]) can be written as f = g^n for g in R, n >= 2. Let P_n := {g^n : g in R} and let mu_n := n Product_{p|n} p. We show among other things that (i) for f in R, f in P_n <=> f mod mu_n in P_n, and (ii) if f in P_n, there...

The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence \(u_{n + d} = a_{1}u_{n + d - 1} + \cdots + a_{d}u_{n}\) holds for all integers \(n\), and every root of \(x^{d} - a_{1}x^{d - 1} - a_{2}x^{d - 2} - \cdots - a_{d}\) is nonzero and simple; then there is no zero term \(u_{n}\) if and only if...