It does not appear that the torsion ambiguity can be eliminated with our present approach, and it is not clear to the author how to modify the arguments for the non-commutative case. We note that our theorem applies in particular to products of non-isogenous elliptic curves. Gajda’s question has its origins in the support problem of P. Erdős: if x and y are positive integers such that for any n...

We apply a heuristic method based on counting points over finite fields to the Poincaré center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain some new information about general degree 3 non linearities. Introduction In 1885 Poincaré asked when the differential equation y = − x+ p(x, y) y + q(x, y) =: − P (x, y) Q(x, ...

We apply a heuristic method based on counting points over finite fields to the Poincaré center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain new evidence for Żo la̧dek’s conjecture about general degree 3 non linearities. Introduction In 1885 Poincaré asked when the differential equation y = − x+ p(x, y) y + q(x, y) =...

We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (−2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo...

Abstract 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 this journal. Subsequently, due general nature underlying problem, we thought that result could have further applications, and be easily overlooked if it appeared appendix. So, with welcome kind help Editors, decided publish se...

We show that the decidability of an amplification of Hilbert’s Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prefixes ∃∀∃ and ∃∃∀∃ are generically decidable. This means, taking the former prefix as an example, that we give a precise geome...

An approach to non-rationality of the curves Y 2 = f(X) using only elementary valuation-theoretic arguments is analyzed and compared with the classical approaches.

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences f(x) ≡ y (mod p) and f(x) ≡ y (mod p), with a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We give two applications of these results to counting hyperelliptic curves in isomorphism classes modulo p and to the diameter of partial trajectories of a...

This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y 2 D dX C Z. We improve the doubling and addition steps in Miller’s algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best res...

We prove, as an analogy of a conjecture of Artin, that if Y −→ X is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides those of Y in Z[T ]. Then, we give some interpretations of this result in terms of semi-abelian varieties.