Browsing by Author "Garcia-Fritz, Natalia"

Now showing 1 - 3 of 3
  • No Thumbnail Available
    Item
    A criterion for nondensity of integral points
    (2024) Garcia-Fritz, Natalia; Pasten, Hector
    We give a general criterion for Zariski degeneration of integral points in the complement of a divisor D$D$ with n$n$ components in a variety of dimension n$n$ defined over Q$\mathbb {Q}$ or over a quadratic imaginary field. The key condition is that the intersection of the components of D$D$ is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest.
  • No Thumbnail Available
    Item
    Families of explicit quasi-hyperbolic and hyperbolic surfaces
    (2020) Garcia-Fritz, Natalia; Urzua, Giancarlo
    We construct explicit families of quasi-hyperbolic and hyperbolic surfaces parametrized by quasi-projective bases. The method we develop in this paper extends earlier works of Vojta and the first author for smooth surfaces to the case of singular surfaces, through the use of ramification indices on exceptional divisors. The novelty of the method allows us to obtain new results for the surface of cuboids, the generalized surfaces of cuboids, and other explicit families of Diophantine surfaces of general type. In particular, we produce new families of smooth complete intersection surfaces of multidegrees m1, horizontal ellipsis. These families give evidence for [6, Conjecture 0.18] in the case of surfaces.
  • No Thumbnail Available
    Item
    Hilbert's tenth problem for lacunary entire functions of finite order
    (2024) Garcia-Fritz, Natalia; Pasten, Hector
    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 for exponential polynomials of finite order (by Chompitaki, Garcia-Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.