On the reducibility type of trinomials

Say a trinomial x^n+A x^m+B \in \Q[x] has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in \Q[x] of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq ... \leq n_k$. Specifying the reducibility type of a monic polynomial of fixed degree is equivalent to specifying rational points on an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable hope that all its rational points may be described; and techniques are available that may also find all points when the genus is 2. Thus all corresponding reducibility types may be described. These low genus instances are the ones studied in this paper.Comment: to appear in Acta Arithmetic

Points at rational distances from the vertices of certain geometric objects

We consider various problems related to finding points in \Q^{2} and in \Q^{3} which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in \Q^{2}, and a cube or tetrahedron in \Q^{3}.Comment: 23 pages, submitte

On certain diophantine equations of diagonal type

In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\in\{(2,4,8,8), (2,8,4,8)\}$. In the case $(p,q,r,s)=(4,4,4,4)$, we present some new parametric solutions of the equation $x^4-y^4=4(z^4-w^4)$.Comment: 16 pages, revised version will appear in the Journal of Number Theor

Lucas sequences whose 12th or 9th term is a square

Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q) is defined by U_0=0, U_1=1, U_n = P U_{n-1}-Q U_{n-2} for n>1. The sequence {U_n(1,-1)} is the familiar Fibonacci sequence, and it was proved by Cohn that the only perfect square greater than 1 in this sequence is $U_{12}=144$. The question arises, for which parameters P, Q, can U_n(P,Q) be a perfect square? In this paper, we complete recent results of Ribenboim and MacDaniel. Under the only restriction GCD(P,Q)=1 we determine all Lucas sequences {U_n(P,Q)} with U_{12}= square. It turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences {U_n(P,Q) with U_9= square.Comment: 13 page

Multisensory perception of looming and receding objects in human newborns

When newborns leave the enclosed spatial environment of the uterus and arrive in the outside world, they are faced with a new audiovisual environment of dynamic objects, actions and events both close to themselves and further away. One particular challenge concerns matching and making sense of the visual and auditory cues specifying object motion [1-5]. Previous research shows that adults prioritise the integration of auditory and visual information indicating looming (for example [2]) and that rhesus monkeys can integrate multisensory looming, but not receding, audiovisual stimuli [4]. Despite the clear adaptive value of correctly perceiving motion towards or away from the self - for defence against and physical interaction with moving objects - such a perceptual ability would clearly be undermined if newborns were unable to correctly match the auditory and visual cues to such motion. This multisensory perceptual skill has scarcely been studied in human ontogeny. Here we report that newborns only a few hours old are sensitive to matches between changes in visual size and in auditory intensity. This early multisensory competence demonstrates that, rather than being entirely na\uefve to their new audiovisual environment, newborns can make sense of the multisensory cue combinations specifying motion with respect to themselves

On two four term arithmetic progressions with equal product

We investigate when two four-term arithmetic progressions have an equal product of their terms. This is equivalent to studying the (arithmetic) geometry of a non-singular quartic surface. It turns out that there are many polynomial parametrizations of such progressions, and it is likely that there exist polynomial parametrizations of every positive degree. We find all such parametrizations for degrees 1 to 4, and give examples of parametrizations for degrees 5 to 10

The simple non-Lie Malcev algebra as a Lie-Yamaguti algebra

The simple 7-dimensional Malcev algebra $M$ is isomorphic to the irreducible $\mathfrak{sl}(2,\mathbb{C})$-module V(6) with binary product $[x,y] = \alpha(x \wedge y)$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $\alpha\colon \Lambda^2 V(6) \to V(6)$. Combining this with the ternary product $(x,y,z) = \beta(x \wedge y) \cdot z$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism \beta\colon \Lambda^2 V(6) \to V(2) \approx \s gives $M$ the structure of a generalized Lie triple system, or Lie-Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied by this binary-ternary structure.Comment: 20 page