28 research outputs found
Lifting, restricting and sifting integral points on affine homogeneous varieties
In a previous paper {GN2} an effective solution of the lattice point counting
problem in general domains in semisimple S-algebraic groups and affine
symmetric varieties was established. The method relies on the mean ergodic
theorem for the action of G on G/Gamma, and implies uniformity in counting over
families of lattice subgroups admitting a uniform spectral gap. In the present
paper we extend some methods developed in {NS} and use them to establish
several useful consequences of this property, including : Effective upper
bounds on lifting for solutions of congruences in affine homogeneous varieties,
effective upper bounds on the number of integral points on general subvarieties
of semisimple group varieties, effective lower bounds on the number of almost
prime points on symmetric varieties, and effective upper bounds on almost prime
solutions of Linnik-type congruence problems in homogeneous varieties.Comment: Submitte
Reverse Mathematics of Divisibility in Integral Domains
This thesis establishes new results concerning the proof-theoretic strength
of two classic theorems of Ring Theory relating to factorization in integral
domains.
The first theorem asserts that if every irreducible is a prime, then every
element has at most one decomposition into irreducibles; the second states
that well-foundedness of divisibility implies the existence of an irreducible
factorization for each element.
After introductions to the Algebra framework used and Reverse Mathematics,
we show that the first theorem is provable in the base system of
Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to
the system ACA0
Résolution de certaines équations diophantiennes et propriétés de certains polynômes
Dans les quatre premiers chapitres de cette thèse, nous abordons quelques équations diophantiennes et leurs solutions. On démontre que l'équation y 2 = px(Ax2 + 2) n'admet qu'un maximum de six solutions entières où p est nombre premier et A > 1 est entier impair ; on démontre que l'équation Resx P(x), x2 + sx + t = a n'admet qu'un nombre ni de solutions (s, t) pour P un polynôme xe et a un entier autre que zéro ; on résout l'équation Fn−Fm = y a lorsque y ∈ {6, 11, 12} et on trouve une borne pour les solutions de Fn + Fm = y a dans le cas général ; et on démontre que si un nombre su sant d'entiers x consécutifs existent tels que P(x) est sous la forme mq lorsque q ≥ 2 est diviseur de deg P, alors P = Rq pour un certain polynôme R, ce qui nous permet de déduire l'existence d'une in nité de solutions à y q = P(x) à partir d'un nombre ni de telles solutions dans certains cas. Dans les six derniers chapitres, nous abordons plusieurs sujets reliés à la décomposition d'objets algébriques. Parmi les résultats, on présente quelques conditions sous lesquelles un polynôme ne peut pas être exprimé comme une composition de deux polynômes de degré inférieur ; on présente une nouvelle démonstration du théorème Carltiz-Lutz sur les polynômes de permutations ; on étudie la possibilité d'exprimer un polynôme comme une somme composée ou un produit composé de deux autres polynômes de degré inférieur ; on trouve une borne pour un des plus petits nombres premiers qui se décompose dans un corps imaginaire quadratique donné ; et on étudie la possibilité de recouvrir un anneau avec ses sous-anneauxThe rst four chapters of this thesis address some Diophantine equations and their solutions. We prove that the equation y 2 = px(Ax2 + 2) has at most six integer solutions (x, y) for p a prime and A > 1 an odd integer; we prove that the equation Resx P(x), x2 + sx + t = a has only nitely many integer solutions (s, t) for a xed polynomial P and nonzero integer a; we completely solve the equation Fn − Fm = y a for y ∈ {6, 11, 12} and bound the solutions for Fn + Fm = y a in general; and we prove that the existence of su ciently many consecutive integers x such that P(x) is of the form mq for q ≥ 2 dividing deg P implies that Rq for some polynomial R, providing criteria for deducing the existence of in nitely many solutions to y q = P(x) from the existence of nitely many solutions in some cases. In the last six chapters, we address various algebraic decomposition related topics. Among other results, we provide criteria which guarantee a polynomial cannot be written as a composition of two polynomials of smaller degree; we provide a new proof of the Carlitz-Lutz theorem on permutation polynomials; we study the possibility of expressing a polynomial as the composed sum or composed multiplication of two polynomials of smaller degree; we bound from below some of the smallest primes which split in an imaginary quadratic eld; and we study the possibility of covering a ring with its subring
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Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations
New striking analogies between H. Hahn’s fields of generalised series with real coefficients, G. H. Hardy’s field of germs of real valued functions, and J. H. Conway’s field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example