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We construct explicit non-isotrivial families of polynomials over $\mathbb{Q}$ satisfying uniform boundedness for their rational preperiodic points.

One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable. Our main result provides a step towards an answer: For every $\rho\ge 0$, we give a negative solution to the analogue of HTP in the ring of complex entire functions in one variable of growth order at most $\rho$.

We revisit a subexponential bound for the $abc$ conjecture due to the first author, and we establish a variation of it using linear forms in logarithms. As an application, we prove an unconditional subexponential bound towards the $4$-terms $abc$ conjecture under a suitable hypothesis on the size of the variables.

Given a complex projective algebraic variety $X$ we define $ h(X)$ as the largest $n$ such that the $n$-th symmetric power of $X$ is (Brody) hyperbolic. Using Nevanlinna theory for algebroid maps, we give non-trivial lower bounds for $ h(X)$. From an arithmetic point of view, the problem is closely related to the finiteness of algebraic points of bounded degree in varieties over number fields. We provide explicit applications of our results in the case of curves embedded in surfaces and in the case of subvarieties of abelian varieties.

We investigate the behavior of active Brownian particles (ABP) within a temporal complex network framework approach. We focused on the node degree distribution, average path length, and average clustering coefficient across the P\'eclet number and packing fraction region. In the single phase or gas region, particle interactions mirror a random graph, and the average number of unique interactions display a non-monotonic behavior with the packing fraction. As we ventured toward the Motility-Phase Separation (MIPS) frontier, a hybrid distribution with a time-evolving pattern emerged, combining Gaussian and power law components. Moreover, we discovered a double power-law distribution in the phase-separated region, with two characteristic slopes symbolizing the emerging gas and solid regions. Our approach involved various numerical and theoretical analyses to capture the role of the packing fraction and P\'eclet number in the topological properties of the dynamical complex network that arises during the ABP dynamics. These findings provide a robust understanding of a phase transition between two states and how this transition manifests as a random to scale-free behavior.

We combine transcendental methods and the modular approaches to the $ABC$ conjecture to show that the largest prime factor of $n^2+1$ is at least of size $(\log_2 n)^2/\log_3n$ where $\log_k$ is the $k$-th iterate of the logarithm. This gives a substantial improvement on the best available estimates, which are essentially of size $\log_2 n$ going back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on subexpoential bounds for the $ABC$ conjecture, which in a case gives the first improvement on a result by Stewart and Yu dating back over two decades. Central to our approach is the connection between Shimura curves and the $ABC$ conjecture developed by the author.

We give a general criterion for Zariski degeneration of integral points in the complement of a divisor $D$ with $n$ components in a variety of dimension $n$ defined over $\mathbb{Q}$ or over a quadratic imaginary field. The key condition is that the intersection of the components of $D$ is not well-approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a GCD bound for algebraic points in varieties, which can be of independent interest., Comment: References updated

The analogue of Hilbert's tenth problem over $\mathbb{Q}$ asks for an algorithm to decide the existence of rational points in algebraic varieties over this field. This remains as one of the main open problems in the area of undecidability in number theory. Besides the existence of rational points, there is also considerable interest in the problem of effectivity: one asks whether the sought rational points satisfy determined height bounds, often expressed in terms of the height of the coefficients of the equations defining the algebraic varieties under consideration. We show that, in fact, Hilbert's tenth problem over $\mathbb{Q}$ with (finitely many) height comparison conditions is undecidable., Comment: References updated

We prove Szpiro's conjecture for elliptic curves over the rationals having $j$-invariant with denominator of logarithmic size with respect to its numerator., Comment: References updated

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.

In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi_j\colon E_j\to \mathbb{P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi_j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. As it has been observed in the literature, the recent proof of the Uniform Manin-Mumford conjecture implies a full solution of the Bogomolov-Fu-Tschinkel conjecture. In this work we prove a generalization of the Bogomolov-Fu-Tschinkel conjecture where instead of even degree-$2$ maps one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the Uniform Manin-Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields., Comment: References added

Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point $P_0\in{X(K)}$ whose $f$-orbit $\mathcal{O}_f(P_0):=\bigl\{f^n(P):n\in\mathbb{N}\bigr\}$ is Zariski dense, then there are many such points. For example, a weak conclusion would be that $X(K)$ is not the union of finitely many (grand) $f$-orbits, while a strong conclusion would be that any set of representatives for the Zariski dense grand $f$-orbits is Zariski dense. We prove statements of this sort for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces., Comment: 56 pages

We characterize the peculiar velocity field of the local large-scale structure reconstructed from the $2M++$ survey, by treating it as a fluid, extracting the divergence via different approximations over a range pf averaged scales. This reconstructed field is important for cosmology, since it was used to correct the peculiar redshifts of the last SNIA compilation Pantheon+. The results have intriguing implications for the LLSS fluid dynamics and particularly for the ``Tilted Cosmology'' model, although those results have to be taken carefully as the velocity field could contain significant bias due to the reconstruction procedure. Those possible bias and its influence in our results are discussed. Representative values of the apparent deceleration parameter ($\Tilde{q}$) are computed, in order to compare our results with the theoretical predictions of the tilted-universe scenario. We conclude that better velocity field reconstructions are necessary in order to constrain the parameters implied in LLSS research and alternative cosmologies., Comment: Improved discussion about bias in the velocity reconstruction

We study the reconstructed deceleration parameter splitting the data in different redshift bins, fitting both a cosmographic luminosity distance and also assuming a flat $\Lambda$CDM model, using the Pantheon+ sample of type Ia supernova data (SNIA). We observe tensions $\sim 2\sigma-3\sigma$ for different redshift and distance indicators if the full sample is used. However, those tensions disappear when the SNIA at $z<0.008$ are removed. If the data is splitted in 2 hemispheres according to our movement w.r.t CMB, a strange $3.8 \sigma$ tension appears in one of the samples between particular redshift bins. Finally, considering posterior distribution as Gaussian, general linear model prefers a positive slope for $q_0$ across redshift bins opposed to a zero slope expected in a $\Lambda$CDM universe. We discuss possible explanations for our results and the influence of lowest redshift SNIA data in cosmological analysis., Comment: Published version in Physics of the Dark Universe journal

We revisited the problem of describing, on average, a fractal distribution of matter using a Lemaitre-Tolman-Bondi (LTB) solution. Here we study the fractal structure of our local universe having a fractal dimension and a scale transition, ensuring an homogeneous bang-time function. We test our model with the latest type Ia supernova data, the Pantheon compilation, and discuss problems and possible improvements for it, concluding that a fractal transition in LTB cosmology cannot be used to explain the effects of dark energy without requiring an inhomogeneous big-bang, but it can be useful to study structures at low scales., Comment: We show that following 10.48550/ARXIV.2302.04679, the model proposed requires an inhomogeneous big bang , therefore not solving the problem of DE. Our model construct a fractal matter model that assumes a homogeneous big bang, not requiring any new physics. We added a different smooth mass function to compare with the sharp one, not changing our principal results

For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes., Comment: 6 pages

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 contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine.

It is an old problem in the area of Diophantine definability to determine whether $\mathbb{Q}$ is Diophantine in $\mathbb{Q}(z)$. We provide a positive answer conditional on two standard conjectures on elliptic surfaces.

A negative solution to Hilbert's tenth problem for the ring of integers $O_F$ of a number field $F$ would follow if $\mathbb{Z}$ were Diophantine in $O_F$. Denef and Lipshitz conjectured that the latter occurs for every number field $F$. In this note we show that the conjecture of Denef and Lipshitz is a consequence of a well-known conjecture on elliptic surfaces. -- Es sabido que se obtendr\'ia una soluci\'on negativa al d\'ecimo problema de Hilbert para el anillo de enteros $O_F$ de un campo de n\'umeros $F$ si $\mathbb{Z}$ fuera diofantino en $O_F$. Denef y Lipshitz conjeturaron que esto \'ultimo ocurre para todo $F$. En esta nota se demuestra que la conjetura de Denef y Lipshitz es consecuencia de una conocida conjetura sobre superficies el\'ipticas., Comment: In Spanish

We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Furthermore, we show that the Lang-Waldschmidt conjecture implies a special case of Vojta's conjecture with truncation in arbitrary dimension. Our methods are based on the theory of linear forms in logarithms and a geometric construction., Comment: Updated references

{We focus on characterizing the high-energy emission mechanisms of blazars by analyzing the variability in the radio band of the light curves of more than a thousand sources. We are interested in assigning complexity parameters to these sources, modeling the time series of the light curves with the method of the Horizontal Visibility Graph (HVG), which allows us to obtain properties from degree distributions, such as a characteristic exponent to describe its stochasticity and the Kullback-Leibler Divergence (KLD), presenting a new perspective to the methods commonly used to study Active Galactic Nuclei (AGN). We contrast these parameters with the excess variance, an astronomical measurement of variability in light curves, at the same time we use the spectral classification of the sources. While it is not possible to find significant correlations with the excess variance, the degree distributions extracted from the network are detecting differences related to the spectral classification of blazars. These differences suggest a chaotic behavior in the time series for the BL Lac sources and a correlated stochastic behavior in the time series for the FSRQ sources. Our results show that complex networks may be a valuable alternative tool to study AGNs according to the variability of their energy output., Comment: 12 pages, 8 figures, 2 tables. Accepted for publication in Entropy