Geometriske og dynamiske egenskapar av rasjonelle funksjonar (avslutta)

• Truong, Trung Tuyen (2026). On the properness of polynomial selfmaps. Vietnam Journal of Mathematics. ISSN 2305-221X. doi: 10.1007/s10013-026-00804-y. Fulltekst i vitenarkiv
• Charles, Favre; Truong, Trung Tuyen & Junyi, Xie (2026). Topological entropy of a rational map over a complete metrized field. Annales Scientifiques de l'Ecole Normale Supérieure. ISSN 0012-9593. 59(1), s. 67–124. doi: 10.24033/asens.2636. Fulltekst i vitenarkiv Vis sammendrag

  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. This extends 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 ϵ-reduction of a Berkovich analytic space, a notion of independent interest.

• Hu, Fei & Truong, Trung Tuyen (2025). A dynamical approach to generalized Weil’s Riemann hypothesis and semisimplicity. Science China Mathematics. ISSN 1674-7283. doi: 10.1007/s11425-024-2363-9. Fulltekst i vitenarkiv Vis sammendrag

  Let f be a dynamical correspondence of a smooth projective variety X over an algebraically closed field of arbitrary characteristic. In 2016, Truong conjectured that the dynamical degrees of f defined via the pullback actions on ℓ-adic 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 particular case. In this paper, we introduce a quantitative strengthening of the standard conjecture C, named as Conjecture Gr, and show that it holds on abelian varieties and Kummer surfaces. We prove that Conjecture Gr yields the generalized Weil’s Riemann hypothesis. Moreover, Conjecture Gr plus the standard conjecture D imply the norm comparison conjecture, whose consequences include the DDC conjecture and the generalized semisimplicity conjecture. As applications, we prove the DDC conjecture for algebraically stable dynamical correspondences of abelian varieties and Kummer surfaces, extending Hu’s previous work (2019, 2024) on endomorphisms of abelian varieties. We also obtain a similar comparison result for effective finite correspondences of abelian varieties and prove the generalized semisimplicity conjecture for Kummer surfaces.

• Hu, Fei (2024). EIGENVALUES AND DYNAMICAL DEGREES OF SELF-MAPS ON ABELIAN VARIETIES. Journal of Algebraic Geometry. ISSN 1056-3911. 33(2), s. 265–293. doi: 10.1090/jag/806. Fulltekst i vitenarkiv Vis sammendrag

  Let X be a smooth projective variety over an algebraically closed field, and f : X → X a surjective self-morphism of X. The i-th cohomological dynamical degree χi(f) is defined as the spectral radius of the pullback f∗ on the ´etale cohomology group Hi´et(X, Ql) and the k-th numerical dynamical degree λk(f) as the spectral radius of the pullback f∗ on the vector space Nk(X)R of real algebraic cycles of codimension k on X modulo numerical equivalence. Truong conjectured that χ2k(f) = λk(f) for all 0 ≤ k ≤ dim X as a generalization of Weil’s Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

• Truong, Trung Tuyen (2024). Relations between dynamical degrees, Weil’s Riemann hypothesis and the standard conjectures. Commentarii Mathematici Helvetici. ISSN 0010-2571. 100(1), s. 61–92. doi: 10.4171/CMH/580. Fulltekst i vitenarkiv Vis sammendrag

  Let K be an algebraically closed field, X a smooth projective variety over K and f:X→X a dominant regular morphism. Let Ni(X) be the group of algebraic cycles, of codimension i, modulo numerical equivalence. Let χ(f) be the spectral radius of the pullback f∗: H∗(X,Ql)→H∗(X,Ql) on l-adic cohomology groups, and λ(f) the spectral radius of the pullback f∗:N∗(X)→N ∗(X). We prove in this paper, by using consequences of Deligne’s proof of Weil’s Riemann hypothesis, that χ(f)=λ(f). This answers affirmatively a question posed by Esnault and Srinivas. Consequently, the algebraic entropy logχ(f) of an endomorphism is both a birational invariant and étale invariant. More general results are proven if either K=Fˉp or the Standard Conjecture D holds (this applies specially to Abelian varieties). Among other results in the paper, we show that if some properties of dynamical degrees, known in the case K=C, hold in positive characteristics, then simple proofs of Weil’s Riemann hypothesis follow. More generally, the analogy in positive characteristic of Serre’s famous result on polarized endomorphisms of compact Kähler manifolds also follows.

• Fornæss, John Erik; Hu, Mi; Truong, Trung Tuyen & Watanabe, Takayuki (2024). Backtracking New Q-Newton’s Method, Newton’s Flow, Voronoi’s Diagram and Stochastic Root Finding. Complex Analysis and Operator Theory. ISSN 1661-8254. 18(5). doi: 10.1007/s11785-024-01558-6. Fulltekst i vitenarkiv Vis sammendrag

  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, with BNQN. In general, they look more smooth than that of Newton’s method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton’s flow and Voronoi’s diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton’s method and Random Relaxed Newton’s method.

• Cinzia, Bisi; Jonathan, Hauenstein & Truong, Trung Tuyen (2024). Some interesting birational morphisms of smooth affine quadric 3-folds. Nonlinearity. ISSN 0951-7715. 37(7). doi: 10.1088/1361-6544/ad48f3. Fulltekst i vitenarkiv
• Barth, Viktor Balch & Truong, Trung Tuyen (2024). Images of dominant endomorphisms of affine space. Communications in Algebra. ISSN 0092-7872. doi: 10.1080/00927872.2024.2418374. Fulltekst i vitenarkiv
• Hu, Fei & Jiang, Chen (2023). Polynomial Volume Growth of Quasi-Unipotent Automorphisms of Abelian Varieties (with an Appendix in Collaboration with Chen Jiang). International Mathematics Research Notices (IMRN). ISSN 1073-7928. 2024(8), s. 6374–6399. doi: 10.1093/imrn/rnad170. Fulltekst i vitenarkiv
• Barth, Viktor Balch (2023). Surjective morphisms from affine space to its Zariski open subsets. International Journal of Mathematics. ISSN 0129-167X. 34(12). doi: 10.1142/S0129167X23500751. Fulltekst i vitenarkiv Vis sammendrag

  We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set [Formula: see text], we construct an endomorphism of [Formula: see text] with [Formula: see text] as its image. By Noether’s normalization lemma, these results extend to give surjective maps from any [Formula: see text]-dimensional affine variety [Formula: see text] to [Formula: see text].

• Truong, Trung Tuyen; To, Tat Dat; Nguyen, Hang-Tuan; Nguyen, Thu Hang; Nguyen, Hoang Phuong & Helmy, Maged (2023). A fast and simple modification of Newton's method avoiding saddle points. Journal of Optimization Theory and Applications. ISSN 0022-3239. 199, s. 805–830. doi: 10.1007/s10957-023-02270-9. Fulltekst i vitenarkiv Vis sammendrag

  We propose in this paper New Q-Newton’s method. The update rule is conceptually very simple, using the projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of the Hessian. The main result of this paper roughly says that if a sequence {xn} constructed by the method from a random initial point x0 converges, then the limit point is a critical point and not a saddle point, and the convergence rate is the same as that of Newton’s method. A subsequent work has recently been successful incorporating Backtracking line search to New Q-Newton’s method, thus resolving the global convergence issue observed for some (non-smooth) functions. An application to quickly find zeros of a univariate meromorphic function is discussed, accompanied with an illustration on basins of attraction.

• Truong, Tuyen Trung & Nguyen, Hang-Tuan (2022). Backtracking gradient descent method and some applications in Large scale optimisation. Part 1: Theory. Minimax Theory and its Applications. ISSN 2199-1413. 7(1), s. 79–108. Fulltekst i vitenarkiv Vis sammendrag

  Deep Neural Networks (DNN) are essential in many realistic applications, including Data Science. At the core of DNN is numerical optimisation, in particular gradient descent methods (GD). The purpose of this paper is twofold. First, we prove some new results on the backtracking variant of GD under very general situations. Second, we present a comprehensive comparison of our new results to the previously known results in the literature, providing pros and cons of these methods. To illustrate the efficiency of Backtracking line search, we will present some experimental results (on validation accuracy, training time and so on) on CIFAR10, based on implemetations developed in another paper by the authors. Source codes for the experiments are available on GitHub.

• Hu, Fei & Truong, Trung Tuyen (2022). An inequality on polarized endomorphisms. Archiv der Mathematik. ISSN 0003-889X. 120, s. 27–34. doi: 10.1007/s00013-022-01802-5. Fulltekst i vitenarkiv
• Truong, Tuyen Trung (2020). Some new theoretical and computational results around the Jacobian conjecture. International Journal of Mathematics. ISSN 0129-167X. 31(7). doi: 10.1142/S0129167X20500500. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we study a so-called Condition C1 on square matrices with complex coefficients and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. We show that these conditions satisfy many good properties and in particular are satisfied by a dense subset of the set of square matrices of a given rank [Formula: see text]. Based on this, we propose a heuristic argument for the truth of the Jacobian conjecture. We propose some new equivalent formulations and some generalizations of the Jacobian conjecture, and some approaches (including computer algebra and numerical methods) toward resolving it. We show that some of these equivalent formulations are automatically satisfied by generic Druzkowski matrices. Applications and experimental results are included.

• Truong, Tuyen Trung (2020). Strong submeasures and applications to non-compact dynamical systems. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. doi: 10.1017/etds.2020.132. Fulltekst i vitenarkiv
• Truong, Tuyen Trung & Nguyen, Hang-Tuan (2020). Backtracking Gradient Descent Method and Some Applications in Large Scale Optimisation. Part 2: Algorithms and Experiments. Applied Mathematics and Optimization. ISSN 0095-4616. doi: 10.1007/s00245-020-09718-8. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we provide new results and algorithms (including backtracking versions of Nesterov accelerated gradient and Momentum) which are more applicable to large scale optimisation as in Deep Neural Networks. We also demonstrate that Backtracking Gradient Descent (Backtracking GD) can obtain good upper bound estimates for local Lipschitz constants for the gradient, and that the convergence rate of Backtracking GD is similar to that in classical work of Armijo. Experiments with datasets CIFAR10 and CIFAR100 on various popular architectures verify a heuristic argument that Backtracking GD stabilises to a finite union of sequences constructed from Standard GD for the mini-batch practice, and show that our new algorithms (while automatically fine tuning learning rates) perform better than current state-of-the-art methods such as Adam, Adagrad, Adadelta, RMSProp, Momentum and Nesterov accelerated gradient. To help readers avoiding the confusion between heuristics and more rigorously justified algorithms, we also provide a review of the current state of convergence results for gradient descent methods. Accompanying source codes are available on GitHub.

• Truong, Tuyen Trung (2020). When will a sequence of points in a Riemannian submanifold converge? Mathematics. 8(11). doi: 10.3390/math8111934. Fulltekst i vitenarkiv Vis sammendrag

  Let X be a Riemannian manifold and xn a sequence of points in X. Assume that we know a priori some properties of the set A of cluster points of xn. The question is under what conditions that xn will converge. An answer to this question serves to understand the convergence behaviour for iterative algorithms for (constrained) optimisation problems, with many applications such as in Deep Learning. We will explore this question, and show by some examples that having X a submanifold (more generally, a metric subspace) of a good Riemannian manifold (even in infinite dimensions) can greatly help.

Sjå alle arbeida i NVA

• Truong, Trung Tuyen (2024). Maths, Optimization and Deep Learning/Machine Learning. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). Applications of mathematics in Machine Learning and Deep Learning. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). A new equivalence and approach to the Riemann hypothesis, via dynamical systems. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). The role of Mathematics and Optimization in Machine Learning/Deep Learning. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). The Riemann hypothesis and the dynamics of Backtracking New Q-Newton's method. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). A NEW NEWTON-TYPE METHOD AND CONNECTiONS TO SCHRODER THEOREM, VORONOi’S DiAGRAMS, NEWTON’S FLOWS AND THE RiEMANN HYPOTHESiS. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). Backtracking New Q-Newton’s method and connection to the Riemann hypothesis. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). The Riemann hypothesis and dynamics of a root finding method. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). A new equivalence and a new approach to the Riemann hypothesis, via complex analysis and dynamical systems. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). The Riemann hypothesis and dynamics of a root finding algorithm. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2024). How is AI doing with mathematical arguments. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Backtracking New Q-Newton's method with applications in stochastic root finding. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Stochastic dynamical systems coming from root finding: phenomena and questions. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Some interesting research topics relevant to Machine Learning/Deep Learning. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Hvordan finner vi løsninger til polynomer i 1 kompleks variabel? Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). A better algorithm to solve equations than Newton's method. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Optimization algorithms: Theory and large scale applications, intensive course. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). How good can AI do simple logical reasoning and mathematics now? - An update. Fulltekst i vitenarkiv
• Hu, Mi (2023). Interior Dynamics of Fatou Sets. Fulltekst i vitenarkiv
• Hu, Mi (2023). Dynamics Inside Attracting Basins in One Dimension. Fulltekst i vitenarkiv
• Hu, Mi (2023). Dynamics Inside Fatou Sets. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). New Q-Newton's method and Backtracking line search. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2023). Periodic points, dynamical degrees and root finding. Fulltekst i vitenarkiv
• Hu, Fei (2022). An upper bound for polynomial log-volume growth of automorphisms of zero entropy. Fulltekst i vitenarkiv
• Hu, Fei (2022). An upper bound for polynomial log-volume growth of automorphisms of zero entropy. Fulltekst i vitenarkiv
• Hu, Fei (2022). An upper bound for polynomial log-volume growth of automorphisms of zero entropy. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2022). Some interesting birational morphisms of smooth affine quadric 3-folds. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2022). Entropy of maps on non-Archimedean fields. Fulltekst i vitenarkiv
• Barth, Viktor Balch (2022). Surjective algebraic maps onto algebraically subelliptic manifolds. Fulltekst i vitenarkiv
• Hu, Fei (2022). Dynamical degrees of self-maps of abelian varieties in positive characteristic. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2022). Entropy of maps on non-Archimedean fields. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2022). Optimization, Lojasiewicz inequalities, Dynamical Systems and Deep Neural Networks (Or: Is there a cat or not a cat?). Fulltekst i vitenarkiv
• Hu, Fei (2022). An upper bound for polynomial log-volume growth of automorphisms of zero entropy. Fulltekst i vitenarkiv
• Hu, Fei (2022). An upper bound for polynomial log-volume growth of automorphisms of zero entropy. Fulltekst i vitenarkiv
• Hu, Fei (2021). Comparison Problems on Dynamical Degrees, Norms and Spectral Radii of Correspondences. Fulltekst i vitenarkiv
• Hu, Fei (2021). A dynamical approach to generalized Weil's Riemann hypothesis. Fulltekst i vitenarkiv
• Hu, Fei (2021). A DYNAMICAL APPROACH TO GENERALIZED WEIL’S RIEMANN HYPOTHESIS. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2021). A 10-line proof of a generalisation of Weil’s RH for Abelian varieties. Fulltekst i vitenarkiv
• Hu, Fei (2021). A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2021). A quick method to find roots of univariate meromorphic functions. Fulltekst i vitenarkiv
• Hu, Fei (2021). A dynamical approach to Weil’s Riemann hypothesis. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2021). Implementation of Backtracking line search in Deep Neural Networks: theory and practice. Fulltekst i vitenarkiv
• Hu, Fei (2021). A dynamical approach to Weil’s Riemann hypothesis. Fulltekst i vitenarkiv
• Barth, Viktor Balch (2021). Surjective algebraic maps onto algebraically subelliptic manifolds. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2020). Rationality of quotients of Abelian varieties and computer algebra. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2020). Rationality of quotients of Abelian varieties and computer algebra. Fulltekst i vitenarkiv
• Truong, Trung Tuyen (2020). Deep Learning and Automated Proof Checking. Fulltekst i vitenarkiv

Sjå alle arbeida i NVA