5 citations · 6 across the 17 of their papers we have counts for
19 papers
A note on Bremner's conjecture and uniformity
Natalia Garcia-Fritz, Hector Pasten
In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose -coordinates are in arithmetic progression, have lar…
Patterns on elliptic curves beyond Bremner's conjecture
Natalia Garcia-Fritz, Hector Pasten
In the late 1990's, Bremner conjectured that long arithmetic progressions among the -coordinates of rational points of an elliptic curve over should force the r…
Towards Lang--Vojta via Degeneration
Ryan C. Chen, Natalia Garcia-Fritz, Siddharth Mathur +1
Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of -integral points of varieties over number fields , including many cases with geom…
Xeric varieties
Natalia Garcia-Fritz, Hector Pasten
Let be a smooth projective variety over a number field . The Green--Griffiths--Lang conjecture relates the question of finiteness of rational points in to the triviality…
Effective Mordell for curves with enough automorphisms
Natalia Garcia-Fritz, Hector Pasten
We prove a completely explicit and effective upper bound for the Néron--Tate height of rational points of curves of genus at least over number fields, provided that they have e…
Algebroid maps and hyperbolicity of symmetric powers
Natalia Garcia-Fritz, Hector Pasten
Given a complex projective algebraic variety we define as the largest such that the -th symmetric power of is (Brody) hyperbolic. Using Nevanlinna theory for…
A criterion for non-density of integral points
Natalia Garcia-Fritz, Hector Pasten
We give a general criterion for Zariski degeneration of integral points in the complement of a divisor with components in a variety of dimension defined over $\mathbb{Q…
Effectivity for existence of rational points is undecidable
Natalia Garcia-Fritz, Hector Pasten, Xavier Vidaux
The analogue of Hilbert's tenth problem over asks for an algorithm to decide the existence of rational points in algebraic varieties over this field. This remains as o…
Linear -coordinate relations of triples on elliptic curves
Jerson Caro, Natalia Garcia-Fritz
For an elliptic curve defined over the field of complex numbers, we classify all translates of elliptic curves in such that the -coordinates satisfy a lin…
Hilbert's tenth problem for lacunary entire functions of finite order
Natalia Garcia-Fritz, Hector Pasten
In the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and…
Intersecting the torsion of elliptic curves
Natalia Garcia-Fritz, Hector Pasten
In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves and along with even degree- maps having different bran…
Non-Diophantine sets in rings of functions
Natalia Garcia-Fritz, Hector Pasten, Thanases Pheidas
Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contr…
A Diophantine definition of the constants in
Natalia Garcia-Fritz, Hector Pasten
It is an old problem in the area of Diophantine definability to determine whether is Diophantine in . We provide a positive answer conditional on two st…
On the conjectures of Vojta and Campana over function fields with explicit exceptional sets
Natalia Garcia-Fritz
We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the except…
Elliptic curves with long arithmetic progressions have large rank
Natalia Garcia-Fritz, Hector Pasten
For any family of elliptic curves over the rational numbers with fixed -invariant, we prove that the existence of a long sequence of rational points whose -coordinates form a…
Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points
Natalia Garcia-Fritz, Hector Pasten
For a positive proportion of primes and , we prove that is Diophantine in the ring of integers of . This provides a new and e…
Families of explicit quasi-hyperbolic and hyperbolic surfaces
Natalia Garcia-Fritz, Giancarlo Urzúa
We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first a…
Quadratic sequences of powers and Mohanty's Conjecture
Natalia Garcia-Fritz
We prove under the Bombieri-Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequenc…
Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic
Natalia Garcia-Fritz, Hector Pasten
We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteri…