arXiv subjects

Tuyen Trung Truong

Publications and source records attributed to Tuyen Trung Truong.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial corners in finite fields beyond the distinct-degree case

We prove a quantitative polynomial Roth theorem for corners in $\mathbb F_p^2$ for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials $ϕ_1,ϕ_2\in \mathbb \mathbb{F}_p [y]$ are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every $A\subset\mathbb F_p^2$ with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+ϕ_1(y),x_2),\qquad (x_1,x_2+ϕ_2(y)) $$ for some $y\in\mathbb F_p^\times$. This improves the estimate $p^{2-1/16}$ of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map $\widetilde F:W\to\mathbb A^3$ from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An $\ell^2$ matrix estimate adapted to such sets completes the resonant case.

math.CA↗

arXiv2026

On the structure and generic non-Cartesianity of polynomials in product spaces

We develop a general theory of Cartesian and non-Cartesian polynomials on products of complex spaces $\mathbb{C}^{n_1} \times \cdots \times \mathbb{C}^{n_k}$. We prove that, for any fixed degree $d \ge 2$, a (Zariski) generic polynomial is non-Cartesian in a broad range of dimensions, establishing that Cartesian structure is highly exceptional. We further introduce effective sufficient criteria for a polynomial to be non-Cartesian. Moreover, we show that being (non)-Catersian can be decided algorithmically via Gröbner basis methods and quantitative forms of Hilbert's Nullstellensatz. As an application, we connect the non-Cartesian condition to incidence geometry, obtaining sharp intersection bounds and constructing extremal configurations that demonstrate the optimality of these estimates.

math.AG↗

arXiv2025

Standard conjecture D and some conjectures around Weil's Riemann hypothesis

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If Standard conjecture D holds (for algebraic cycles of dimension $=\dim (X)$) on $X\times X$, then all polarised endomorphisms on $X$ are semisimple. Result 2: We provide heuristic arguments to show that Standard Conjecture D should imply both Dynamical degree comparison conjecture (a generalisation of both Weil's Riemann hypothesis and Tate's question on the absolute value of the eigenvalues of polarised endomorphisms), Norm comparison conjecture (allowing to bound the growth of the pullback of iterations of an endomorphism on cohomology groups in terms of that on algebraic cycles, in particularly implying the semisimplicity of polarised endomorphisms), and Conjecture $G_r$ (which together with Standard conjecture D imply the previous mentioned two conjectures), proposed in previous works by Fei Hu and the author. The heuristic argument relies on the possibility of defining the self-composition $Fr_X^s$ in a good way, where $s$ is an arbitrary rational number (allowed to be negative), and similarly for another object related to the Frobenius morphism.

math.DS↗

arXiv2025

Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee

In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties: If $\{x_n\}$ is a sequence constructed by one such algorithm then: - Finding critical points: Any cluster point of $\{x_n\}$ is a critical point of the cost function $f$. - Convergence guarantee: Under suitable assumptions, the sequence $\{x_n\}$ either converges to a point $x^*$, or diverges to $\infty$. - Avoidance of saddle points: If $x_0$ is randomly chosen, then the sequence $\{x_n\}$ cannot converge to a saddle point. Our results apply for quite general situations: the cost function $f$ is assumed to be only $C^2$ or $C^3$, and either $f$ has at most countably many critical points (which is a generic situation) or satisfies certain Lojasiewicz gradient inequalities. To illustrate the results, we provide a nice application with optimisation over the unit sphere in a Euclidean space. As for tools needed for the results, in the Riemannian manifold case we introduce a notion of "strong local retraction" and (to deal with Newton's method type) a notion of "real analytic-like strong local retraction". In the case of Banach spaces, we introduce a slight generalisation of the notion of "shyness", and design a new variant of Backtracking New Q-Newton's method which is more suitable to the infinite dimensional setting (and in the Euclidean setting is simpler than the current versions).

math.OC↗

arXiv2025

Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture

Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds). We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of $K3$ surfaces. Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism $f:X\rightarrow X$, all eigenvalues of the action of $f$ on $H^k(X)$ have the same absolute value.

math.AG↗

arXiv2025

Backtracking New Q-Newton's method for finding roots of meromorphic functions in 1 complex variable: Global convergence, and local stable/unstable curves

In this paper, we research more in depth properties of Backtracking New Q-Newton's method (recently designed by the third author), when used to find roots of meromorphic functions. If $f=P/Q$, where $P$ and $Q$ are polynomials in 1 complex variable z with $°(P)>°(Q)$, we show the existence of an exceptional set $\mathcal{E}\subset\mathbf{C}$, which is contained in a countable union of real analytic curves in $\mathbf{R}^2=\mathbf{C}$, so that the following statements A and B hold. Here, $\{z_n\}$ is the sequence constructed by BNQN with an initial point $z_0$ which is not a pole of $f$. A) If $z_0\in\mathbf{C}\backslash\mathcal{E}$, then $\{z_n\}$ converges to a root of $f$. B) If $z_0\in \mathcal{E}$, then $\{z_n\}$ converges to a critical point - but not a root - of $f$. Experiments seem to indicate that in general, even when $f$ is a polynomial, the set $\mathcal{E}$ is not contained in a finite union of real analytic curves. We provide further results relevant to whether locally $\mathcal{E}$ is contained in a finite number of real analytic curves. A similar result holds for general meromorphic functions. Moreover, unlike previous work, here we do not require that the parameters of BNQN are random, or that the meromorphic function $f$ is generic. Based on the theoretical results, we explain (both rigorously and heuristically) of what observed in experiments with BNQN, in previous works by the authors. In particular, the dynamics of BNQN (an iterative method) seems to have some striking similarities to Newton's method (a continuous method) and the classical Poincaré-Bendixon theorem for differentiable real dynamical systems on the complex plane. This is the more interesting given that discrete versions of Newton's method (e.g. Relaxed Newton's method) does not behave this way.

math.DS↗

arXiv2025

Uniqueness of tangent currents for positive closed currents

Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a Kähler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-smooth domain. Let $T$ be a positive closed currents of bidegree $(p,p)$ in $X$ such that $T$ satisfies a mild reasonable assumption in a neighborhood of $\partial B$ in $X$ and that the $j$-th average mean $ν_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ converges sufficiently fast to the $j$-th generalized Lelong number $ν_j(T,B)$ as $r$ tends to $0$ so that $r^{-1}(ν_j(T, B,r)-ν_j( T,B))$ is locally integrable near $r=0.$ Then we show that $T$ admits a unique tangent current along $B.$ A local version where we replace the condition of $T$ near $B$ by the conditions on a finite cover of $B$ by piecewise $\mathcal{C}^2$-smooth domains in $V$ is also given. When $T$ is a current of integration over a complex analytic set, we show that $ν_j(T,B,r)-ν_j(T,B)=O(r^ρ)$ for some $ρ>0,$ and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case $l=0$ to the case $l>0.$ The result has applications in the intersection theory of positive closed currents.

math.CV↗

arXiv2024

The Riemann hypothesis and dynamics of Backtracking New Q-Newton's method

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - was recently introduced by the second author. This method has good global convergence guarantees, specially concerning finding roots of meromorphic functions. This paper explores using BNQN for the Riemann xi function. We show in particular that the Riemann hypothesis is equivalent to that all attractors of BNQN lie on the critical line. We also explain how an apparent relation between the basins of attraction of BNQN and Voronoi's diagram can be helpful for verifying the Riemann hypothesis or finding a counterexample to it. Some illustrating experimental results are included, which convey some interesting phenomena. The experiments show that BNQN works very stably with highly transcendental functions like the Riemann xi function and its derivatives. Based on insights from the experiments, we discuss some concrete steps on using BNQN towards the Riemann hypothesis, by combining with de Bruijn -Newman's constant. Ideas and results from this paper can be extended to other zeta functions.

math.DS↗

arXiv2024

Some interesting birational morphisms of smooth affine quadric $3$-folds

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are cohomologically hyperbolic, b) their second dynamical degree is an algebraic number but not an algebraic integer, and c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on $\mathbb{C}^4$ preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

math.AG↗

arXiv2024

Creating walls to avoid unwanted points in root finding and optimization

In root finding and optimization, there are many cases where there is a closed set $A$ one likes that the sequence constructed by one's favourite method will not converge to A (here, we do not assume extra properties on $A$ such as being convex or connected). For example, if one wants to find roots, and one chooses initial points in the basin of attraction for 1 root $z^*$ (a fact which one may not know before hand), then one will always end up in that root. In this case, one would like to have a mechanism to avoid this point $z^*$ in the next runs of one's algorithm. Assume that one already has a method IM for optimization (and root finding) for non-constrained optimization. We provide a simple modification IM1 of the method to treat the situation discussed in the previous paragraph. If the method IM has strong theoretical guarantees, then so is IM1. As applications, we prove two theoretical applications: one concerns finding roots of a meromorphic function in an open subset of a Riemann surface, and the other concerns finding local minima of a function in an open subset of a Euclidean space inside it the function has at most countably many critical points. Along the way, we compare with main existing relevant methods in the current literature. We provide several examples in various different settings to illustrate the usefulness of the new approach.

math.OC↗

arXiv2024

Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding

math.OC↗

arXiv2023

Backtracking New Q-Newton's method, Schröder's theorem, and Linear Conjugacy

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions using BNQN. In particular, it seems that for finding roots of polynomials of degree 2, the basins of attraction of the dynamics for BNQN are the same as that for Newton's method (the latter is the classical Schröder's result in Complex Dynamics). In this paper, we show that indeed the picture we obtain when finding roots of polynomials of degree 2 is the same as that in Schöder's result, with a remarkable difference: on the boundary line of the basins, the dynamics of Newton's method is chaotic, while the dynamics of BNQN is more smooth. On the way to proving the result, we show that BNQN (in any dimension) is invariant under conjugation by linear operators of the form $A=cR$, where $R$ is unitary and $c>0$ a constant. This again illustrates the similarity-difference relation between BNQN and Newton's method.

math.DS↗

arXiv2023

Images of dominant endomorphisms of affine space

A basic problem in the study of algebraic morphisms is to determine which sets can be realised as the image of an endomorphism of affine space. This paper extends the results previously obtained by the first author on the question of existence of surjective maps $F\colon \mathbb{A}^n \rightarrow \mathbb{A}^n\setminus Z$, where $Z$ is an algebraic subvariety of $\mathbb{A}^n$ of codimension at least 2. In particular, we show that for any (affine) algebraic variety $Z$ of dimension at most $n-2$, there is an algebraic variety $W\subset \mathbb{A}^n$ birational to $Z$ and a surjective algebraic morphism $\mathbb{A}^n\rightarrow \mathbb{A}^n\setminus W$. We also propose a conjectural approach towards resolving unknown cases.

math.AG↗

arXiv2022

Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations

In this paper, by combining the algorithm New Q-Newton's method - developed in previous joint work of the author - with Armijo's Backtracking line search, we resolve convergence issues encountered by Newton's method (e.g. convergence to a saddle point or having attracting cycles of more than 1 point) while retaining the quick rate of convergence for Newton's method. We also develop a family of such methods, for general second order methods, some of them having the favour of quasi-Newton's methods. The developed algorithms are very easy to implement. From a Dynamical Systems' viewpoint, the new iterative method has an interesting feature: while it is deterministic, its dependence on Armijo's Backtracking line search makes its behave like a random process, and thus helps it to have good performance. On the experimental aspect, we compare the performance of our algorithms with well known variations of Newton's method on some systems of equations (both real and complex variables). We also explore some basins of attraction arising from Backtracking New Q-Newton's method, which seem to be quite regular and do not have the fractal structures as observed for the standard Newton's method. Basins of attraction for Backtracking Gradient Descent seem not be that regular.

math.OC↗

arXiv2022

Topological entropy of a rational map over a complete metrized field

We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the e-reduction of a Berkovich analytic space, a notion of independent interest.

math.DS↗

arXiv2022

CapillaryX: A Software Design Pattern for Analyzing Medical Images in Real-time using Deep Learning

Recent advances in digital imaging, e.g., increased number of pixels captured, have meant that the volume of data to be processed and analyzed from these images has also increased. Deep learning algorithms are state-of-the-art for analyzing such images, given their high accuracy when trained with a large data volume of data. Nevertheless, such analysis requires considerable computational power, making such algorithms time- and resource-demanding. Such high demands can be met by using third-party cloud service providers. However, analyzing medical images using such services raises several legal and privacy challenges and does not necessarily provide real-time results. This paper provides a computing architecture that locally and in parallel can analyze medical images in real-time using deep learning thus avoiding the legal and privacy challenges stemming from uploading data to a third-party cloud provider. To make local image processing efficient on modern multi-core processors, we utilize parallel execution to offset the resource-intensive demands of deep neural networks. We focus on a specific medical-industrial case study, namely the quantifying of blood vessels in microcirculation images for which we have developed a working system. It is currently used in an industrial, clinical research setting as part of an e-health application. Our results show that our system is approximately 78% faster than its serial system counterpart and 12% faster than a master-slave parallel system architecture.

eess.IV↗

arXiv2022

CapillaryNet: An Automated System to Quantify Skin Capillary Density and Red Blood Cell Velocity from Handheld Vital Microscopy

Capillaries are the smallest vessels in the body responsible for delivering oxygen and nutrients to surrounding cells. Various life-threatening diseases are known to alter the density of healthy capillaries and the flow velocity of erythrocytes within the capillaries. In previous studies, capillary density and flow velocity were manually assessed by trained specialists. However, manual analysis of a standard 20-second microvascular video requires 20 minutes on average and necessitates extensive training. Thus, manual analysis has been reported to hinder the application of microvascular microscopy in a clinical environment. To address this problem, this paper presents a fully automated state-of-the-art system to quantify skin nutritive capillary density and red blood cell velocity captured by handheld-based microscopy videos. The proposed method combines the speed of traditional computer vision algorithms with the accuracy of convolutional neural networks to enable clinical capillary analysis. The results show that the proposed system fully automates capillary detection with an accuracy exceeding that of trained analysts and measures several novel microvascular parameters that had eluded quantification thus far, namely, capillary hematocrit and intracapillary flow velocity heterogeneity. The proposed end-to-end system, named CapillaryNet, can detect capillaries at $\sim$0.9 seconds per frame with $\sim$93\% accuracy. The system is currently being used as a clinical research product in a larger e-health application to analyse capillary data captured from patients suffering from COVID-19, pancreatitis, and acute heart diseases. CapillaryNet narrows the gap between the analysis of microcirculation images in a clinical environment and state-of-the-art systems.

eess.IV↗

arXiv2021

A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity

Let $X$ be a smooth projective variety over an algebraically closed field of arbitrary characteristic, and $f$ a dynamical correspondence of $X$. In 2016, the second author conjectured that the dynamical degrees of $f$ defined by the pullback actions on étale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil's Riemann hypothesis (for polarized endomorphisms) as a special case. To proceed, we introduce the so-called Conjecture $G_r$, which is a quantitative strengthening of the standard conjecture $C$ and holds on abelian varieties and Kummer surfaces. We prove that for arbitrary varieties, Conjecture $G_r$ yields the generalized Weil's Riemann hypothesis. Moreover, Conjecture $G_r$ plus the standard conjecture $D$ imply the so-called norm comparison (NC) conjecture, whose consequences include the DDC conjecture and the generalized semisimplicity conjecture (for polarized endomorphisms). As an application, we obtain new results on the DDC conjecture for abelian varieties and Kummer surfaces, and the generalized semisimplicity conjecture for Kummer surfaces. Finally, we also obtain a similar comparison result for effective finite correspondences of abelian varieties.

math.AG↗