Factors of disconnected graphs and polynomials with nonnegative integer coefficients

We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has $n$ connected components, where $n$ is prime, or $n=1,4,8,9$, and satisfies some additional conditions, it factors uniquely under the given products. If, on the contrary, $n=6$ or 10, all cases of nonunique factorisation are described precisely.Comment: 14 page

mathematische

The well known binary, decimal,..., number systems in the integers admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b | ≥ 4, and for b = −2. For the special case b = −2, we give a full characterisation of all valid digit sets. Key words: Radix systems 1 Introduction an

The Casas-Alvero conjecture for infinitely many degrees

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been some computational verifications for small degrees d. In this paper the conjecture is proved in the case where the degree of the polynomial is a power of a prime number, or twice such a power. Moreover, for each positive characteristic p, we give an example of a polynomial of degree d which is not a dth power but which has a common factor with each of its first d-1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.Comment: 7 pages; v2: corrected some typos and references, and added section on computational aspect

The Cartesian product of graphs with loops

We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.Comment: 8 pages, 1 figur

Deterministic Equation Solving over Finite Fields

Deterministic algorithms are presented for the efficient solution of diagonal homogeneous equations in many variables over finite fields. As auxiliary algorithms, it is shown how to compute a field generator that is an nth power, and how to write elements as sums of nth powers, for a given integer n. All these algorithms take polynomial time in n and in the logarithm of the field size, and are practical as stated

Construction of Rational Points on Elliptic Curves over Finite Fields

We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero