Composite lacunary polynomials and the proof of a conjecture of Schinzel

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then h(x)\in \C[x] must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by Erd\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an "algorithmic" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\Z^l$.Comment: 9 page

Integral points on curves f(X)-f(Y)X-Y

The following short article first arose as an Appendix to the paper Counting points of bounded height in monoid orbits, by Wade Hindes, which appears just above in this journal. Subsequently, due to the general nature of the underlying problem, we thought that the result could have further applications, and could be easily overlooked if it appeared as an appendix. So, with the welcome kind help of the Editors, we decided to publish the result separatel

Hyperelliptic continued fractions and generalized jacobians

For a complex polynomial D(t) of even degree, one may define the continued fraction of D(t). This was found relevant already by Abel in 1826, and then by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange\u2019s theorem still exists in full generality: pre-periodicity of the degrees of the partial quotients always holds. Apparently, this fact was never noted before. This also yields a corresponding formula for the degrees of the convergents, for which we shall prove new bounds which are generally best possible (halving the known ones). We shall further study other aspects of the continued fraction, like the growth of the heights of partial quotients. Throughout, some striking phenomena appear, related to the geometry of (gen-eralized) Hyperelliptic Jacobians. Another conclusion central in this paper concerns the poles of the convergents: there can be only finitely many rational ones which occur infinitely many times. (This is crucial for certain applications to a function field version of a question of McMullen.) Our methods rely, among other things, on linking Pad\ue9 approximants and convergents with divisor relations in generalized Jacobians; this shall allow an application of a version for algebraic groups, proved in this paper, of the Skolem-Mahler-Lech theorem

Bounded Height in Pencils of Finitely Generated Subgroups

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques

Torsion points on families of squares of elliptic curves

In a recent paper we proved that there are at most finitely many complex numbers λ ≠ 0,1 such that the points $${(2,\sqrt{2(2-\lambda)})}$$ and $${(3, \sqrt{6(3-\lambda)})}$$ are both torsion on the elliptic curve defined by Y 2=X(X − 1)(X − λ). Here we give a generalization to any two points with coordinates algebraic over the field Q(λ) and even over C(λ). This implies a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic scheme

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

We offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in (Formula presented.), there are only finitely many directions for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori

On some notions of good reduction for endomorphisms of the projective line

Let $\Phi$ be an endomorphism of \SR(\bar{\Q}), the projective line over the algebraic closure of \Q, of degree $\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $\Phi$ has critically good reduction at $v$ if any pair of distinct ramification points of $\Phi$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $\Phi$ has simple good reduction at $v$ if the map $\Phi_v$, the reduction of $\Phi$ modulo $v$, has the same degree of $\Phi$. We prove that if $\Phi$ has critically good reduction at $v$ and the reduction map $\Phi_v$ is separable, then $\Phi$ has simple good reduction at $v$.Comment: 15 page

Composite factors of binomials and linear systems in roots of unity

In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g(h(x)) for polynomials g, h both of degree > 1. In particular, we prove that, if a binomial has such a composite factor, then deg g 64 2 (under natural necessary conditions). This is best-possible and improves on a previous bound deg g 64 24. This result provides evidence toward a conjecture predicting a similar bound when binomials are replaced by polynomials with any given number of terms. As an auxiliary result, which could have other applications, we completely classify the solutions in roots of unity of certain systems of linear equations

Diophantine equations with power sums and Universal Hilbert Sets

We study the diophantine equation $f(u(n),y)=0$, where $f(x,y)$ is a polynomial with integral coefficients and $u:N\to Z$ is a sequence expressed as a power sum with integral bases. We completely classify the cases with infinitely many solutions. We also solve the divisibility problem of deciding when can the values of such a power sum divide infinitely often the values of another power sum