194 research outputs found
Composite lacunary polynomials and the proof of a conjecture of Schinzel
Let be a fixed non-constant complex polynomial. It was conjectured by
Schinzel that if 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 ;
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 , where is a polynomial with a
given number of terms and are arbitrary polynomials. As a corollary, this
implies for instance that a polynomial with 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 .Comment: 9 page
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 and 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
On some notions of good reduction for endomorphisms of the projective line
Let be an endomorphism of \SR(\bar{\Q}), the projective line over
the algebraic closure of \Q, of degree defined over a number field
. Let be a non-archimedean valuation of . We say that has
critically good reduction at if any pair of distinct ramification points of
do not collide under reduction modulo and the same holds for any
pair of branch points. We say that has simple good reduction at if
the map , the reduction of modulo , has the same degree of
. We prove that if has critically good reduction at and the
reduction map is separable, then has simple good reduction at
.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 , where is a polynomial with integral coefficients and 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
Anomalous Subvarieties—Structure Theorems and Applications
When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about the boundedness of absolute height on such intersections as well as their finitenes
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