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An algorithm for two‐variable rational interpolation is developed. The algorithm is suitable for interpolation cases where neither the number of interpolation points nor the final degrees of the rational interpolant are known a priori. Instead, a maximum degree for the interpolant's numerator and denominator is assumed. By testing the condition number of the interpolation system's matrix at each step, the necessary reductions are made in order to cope with nonnormality and unattainability occasions. The algorithm can be used for applications of the Evaluation–Interpolation technique in matrix manipulations, such as finding the inverse of a matrix with elements rational functions of two variables. Symbolic calculations are completely avoided, thus keeping the execution time very low even if the system size is large. Most importantly, the algorithm achieves accurate function recoveries for greater polynomial degrees than other bivariate rational interpolation methods. [ABSTRACT FROM AUTHOR]

Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle 0 < θ < π , the number of perfect triangles with an angle θ is finite. [ABSTRACT FROM AUTHOR]

For positive rank r elliptic curves E (Q) , we employ ideal class pairings E (Q) × E − D (Q) → CL (− D) , for quadratic twists E − D (Q) with a suitable "small y -height" rational point, to obtain explicit class number lower bounds that improve on earlier work by the authors. For the curves E (a) : y 2 = x 3 − a , with rank r (a) , this gives h (− D) ≥ 1 10 ⋅ | E tor (Q) | R Q (E) ⋅ π r (a) 2 2 r (a) Γ (r (a) 2 + 1) ⋅ log ⁡ (D) r (a) 2 log ⁡ log ⁡ D , representing a general improvement to the classical lower bound of Goldfeld, Gross and Zagier when r (a) ≥ 3. We prove that the number of twists E − D (a) (Q) with such a suitable point (resp. with such a point and rank ≥2 under the Parity Conjecture) is ≫ a , ε X 1 2 − ε. We give infinitely many cases where r (a) ≥ 6. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank ≥2 quadratic twists, where in addition we obtain "log-power" improvements to the Goldfeld-Gross-Zagier class number lower bound. [ABSTRACT FROM AUTHOR]

We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (xj). Furthermore, the sequence of local maximum errors per interval (xj− 1,xj) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails. [ABSTRACT FROM AUTHOR]

Let X (resp. Y) be a curve of genus 1 (resp. 2) over a base field k whose characteristic does not equal 2. We give criteria for the existence of a curve Z over k whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of X and Y. Moreover, we give algorithms to construct the curve Z once equations for X and Y are given. The first of these is based on interpolation methods involving numerical results over C that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of Y whose desingularization is isomorphic to X. As an application, we find a twist of a Jacobian over Q that admits a rational 70-torsion point. [ABSTRACT FROM AUTHOR]

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah [3, Theorem 1.1]. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface. [ABSTRACT FROM AUTHOR]

The Arakelov-Zhang pairing < ψ,φ > is a measure of the "dynamical distance" between two rational maps ψ and φ defined over a number field K. It is defined in terms of local integrals on Berkovich space at each completion of K. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height hφ and the standard Weil height h, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction. [ABSTRACT FROM AUTHOR]

The Pila–Wilkie theorem states that if a set |$X\subseteq \mathbb{R}^n$| is definable in an o-minimal structure |$\mathcal{R}$| and contains 'many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion |$\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$| of |${\mathcal{R}}$| by a dense set P , which is either an elementary substructure of |${\mathcal{R}}$|⁠ , or it is |$\mathrm{dcl}$| -independent, as follows. If X is definable in |$\widetilde{\mathcal{R}}$| and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is |${\emptyset}$| -definable in |$\langle \overline{\mathbb{R}}, P\rangle$|⁠ , where |$\overline{\mathbb{R}}$| is the real field. Along the way we introduce the notion of the 'algebraic trace part' |$X^{{\, alg}}_t$| of any set |$X\subseteq \mathbb{R}^n$|⁠ , and we show that if X is definable in an o-minimal structure, then |$X^{{\, alg}}_t$| coincides with the usual algebraic part of X. [ABSTRACT FROM AUTHOR]

In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f ,  g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every n ≥ 2 in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square. [ABSTRACT FROM AUTHOR]

We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve |$X/\mathbb{Q}$| whose Jacobian has Mordell–Weil rank larger than its genus. [ABSTRACT FROM AUTHOR]

Let [K: Q] = p be a prime number and let E/K be an elliptic curve with j(E) ∈ Q. We determine the all possibilities for E(K)tors. We obtain these results by studying Galois representations of E and of its quadratic twists. [ABSTRACT FROM AUTHOR]

For a number field |$K$|⁠ , an algebraic variety |$X/K$| is said to have the Hilbert Property if |$X(K)$| is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a |$K$| -rational point in |${\mathbb{P}}_n/K$|⁠ , for |$n \geq 3$|⁠. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove some Brosowski-Meinardus type invariant point results for the set of ε-simultaneous approximation and ε-simultaneous coapproximation for rational type contraction mappings defined on Takahashi spaces. Subsequently, we deduce some results on ε-approximation, ε-coapproximation, best approximation and best coapproximation for such class of mappings. [ABSTRACT FROM AUTHOR]

In [2] , the author claims that the fields Q (D 4 ∞) defined in the paper and the compositum of all D 4 extensions of Q coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of E defined over Q when base changed to the compositum of all D 4 extensions of Q main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal. [ABSTRACT FROM AUTHOR]

In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón–Bezerra–Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández. [ABSTRACT FROM AUTHOR]

We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic. [ABSTRACT FROM AUTHOR]

This is a correction to: Francesca Balestrieri, Rachel Newton, Arithmetic of Rational Points and Zero-cycles on Products of Kummer Varieties and K3 Surfaces, International Mathematics Research Notices, Volume 2021, Issue 6, March 2021, Pages 4255–4279, https://doi.org/10.1093/imrn/rny303In the originally published version of this manuscript, a wrong grant reference was stated in error. This should read: "EPSRC grant EP/S004696/1" instead of: "EPSRC grant (EP/R021422/1)".The grant details have been emended only within this correction notice to preserve the published version of record. [Extracted from the article]

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than a specified bound, where "non-special curve" means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic 3. [ABSTRACT FROM AUTHOR]

In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines are the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Szőnyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size as a unital and meet affine lines of PG (2 , q 2) in one of 4 possible intersection numbers, each of them congruent to 1 modulus q. As a byproduct, we also determine the intersection sizes of the Hermitian curve defined over GF (q 2) , q a square, with suitable rational curves of degree q and we obtain q -divisible codes with 5 non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions up to some q -powers. [ABSTRACT FROM AUTHOR]

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve. [ABSTRACT FROM AUTHOR]

The Kawaguchi–Silverman conjecture predicts that if |$f: X \dashrightarrow X$| is a dominant rational-self map of a projective variety over |$\overline{{\mathbb{Q}}}$|⁠ , and |$P$| is a |$\overline{{\mathbb{Q}}}$| -point of |$X$| with a Zariski dense orbit, then the dynamical and arithmetic degrees of |$f$| coincide: |$\lambda _1(f) = \alpha _f(P)$|⁠. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than |$1$|⁠ , and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism |$f: X \to X$| of a hyper-Kähler manifold defined over |$\overline{{\mathbb{Q}}}$|⁠. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity. [ABSTRACT FROM AUTHOR]

For any family of elliptic curves over the rational numbers with fixed |$j$| -invariant, we prove that the existence of a long sequence of rational points whose |$x$| -coordinates form a nontrivial arithmetic progression implies that the Mordell–Weil rank is large, and similarly for |$y$| -coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings. [ABSTRACT FROM AUTHOR]

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article. [ABSTRACT FROM AUTHOR]

A Ceva point of a rational-sided triangle is any internal or external point such that the lengths of the three cevians through this point are rational. Buchholz [Buc89] studied Ceva points and showed a method to construct new Ceva points from a known one. We prove that almost-equilateral and equilateral rational triangles have infinitely many Ceva points by establishing a correspondence to points in certain elliptic surfaces of positive rank. [ABSTRACT FROM AUTHOR]

Given a smooth projective variety X over a number field k and P ∈ X(k), the first author conjectured that in a precise sense, any sequence that approximates P sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if X is a Q-factorial terminal split toric variety of arbitrary dimension, then P is better approximated by points on a rational curve than by any Zariski dense sequence. [ABSTRACT FROM AUTHOR]

We provide a geometric construction of [ n , 9 , n - 9 ] q near-MDS codes arising from elliptic curves with n F q -rational points. Furthermore, we show that in some cases these codes cannot be extended to longer near-MDS codes. [ABSTRACT FROM AUTHOR]

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of Fq2-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q. [ABSTRACT FROM AUTHOR]

Let |$k$| be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of |$k$| to that of zero-cycles over |$k$| for Kummer varieties over |$k$|⁠. For example, for any Kummer variety |$X$| over |$k$|⁠ , we show that if the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on |$X$| over all finite extensions of |$k$|⁠ , then the (⁠|$2$| -primary) Brauer–Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on |$X$| over |$k$|⁠. We also obtain similar results for products of Kummer varieties, K3 surfaces, and rationally connected varieties. [ABSTRACT FROM AUTHOR]

Given a scheme X over Z and a hyperfield H which is equipped with a topology which satisfies certain conditions, we endow the set X (H) of H -rational points with a natural topology. We then prove that; (1) when H is the Krasner hyperfield , X (H) is homeomorphic to the underlying space of X , (2) when H is the tropical hyperfield and X is of finite type over a complete non-Archimedean valued field k , X (H) is homeomorphic to the underlying space of the Berkovich analytification X an of X , and (3) when H is the hyperfield of signs , X (H) is homeomorphic to the underlying space of the real scheme X r associated with X. [ABSTRACT FROM AUTHOR]

The major objective of this article is to determine and formulate the analytical solutions of the following systems of rational recursive equations: xn+1 = xn-1yn-3/yn-1 (±1 ≠ xn-1yn-3), yn+1 = yn-1xn-3/xn-1 (≠1 ± yn-1 xn-3), n = 0, 1, ..., where the initial conditions x-3; x-2; x-1; x0; y-3; y-2; y-1 and y0 are required to be arbitrary non-zero real numbers. We also introduce some graphs describing these exact solutions under a suitable choice of some initial conditions. [ABSTRACT FROM AUTHOR]

For p an odd prime number, q_{0}=p^{t}, and q=p^{2t-1}, let \mathcal {X}_{\mathcal {G}_{\mathcal {S}}} be the nonsingular model of \displaystyle Y^{q}-Y=X^{q_{0}}(X^{q}-X). In the present work, the number of \mathbb{F}_{q^{n}}-rational points and the full automorphism group of \mathcal {X}_{\mathcal {G}_{\mathcal {S}}} are determined. In addition, the L-polynomial of this curve is provided, and the number of \mathbb{F}_{q^{n}}-rational points on the Jacobian J_{\mathcal {X}_{\mathcal {G}_{\mathcal {S}}}} is used to construct étale covers of \mathcal {X}_{\mathcal {G}_{\mathcal {S}}}, some with many rational points. [ABSTRACT FROM AUTHOR]

Let Z be a quadratic hypersurface of Pn(R) defined over Q containing points whose coordinates are linearly independent over Q. We show that, among these points, the largest exponent of uniform rational approximation is the inverse 1/ρ of an explicit Pisot number ρ<2 depending only on n if the Witt index (over Q) of the quadratic form q defining Z is at most 1, and that it is equal to 1 otherwise. Furthermore, there are points of Z which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound 1/ρ uses a recent transference inequality of Marnat and Moshchevitin. In the case n=2, we recover results of the second author while for n>2, this completes recent work of Kleinbock and Moshchevitin. [ABSTRACT FROM AUTHOR]

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of F q 2 -rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point. [ABSTRACT FROM AUTHOR]

Consider a one-parameter family of smooth, irreducible, projective curves of genus |$g\ge 2$| defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family. [ABSTRACT FROM AUTHOR]

This note reformulates Mazur's result on the possible orders of rational torsion points on elliptic curves over Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of 'pointless' genus one curves over Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur's theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry. [ABSTRACT FROM AUTHOR]

In 1922 Mordell conjectured the striking statement that, for a polynomial equation ƒ(x,y) = 0, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983 and again by a different method by Vojta in 1991. But neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of p-adic Galois representations; this is the subject of the present exposition. [ABSTRACT FROM AUTHOR]

We exhibit some new infinite families of rational values of τ, some of them squares of rationals, for which the group or even the semigroup generated by the matrices (¹0¹1 ) and ( ¹τ01 ) is not free. [ABSTRACT FROM AUTHOR]

We re-prove the classification of motions of an octahedron — obtained by Bricard at the beginning of the XX century — by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic. [ABSTRACT FROM AUTHOR]

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain θ(x1, y1, z1) = θ(x2, y2, z2) = θ(x3, y3, z3) = k where θ(x, y, z) is a sextic form defined by θ(x, y, z) = x6 +y6 +z6 -2x³y³ -2x³z³ -2y³z³ and k is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve y² = x³ + k=4. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parametrized family of Mordell curves of generic rank ≥ 6 using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is 6. [ABSTRACT FROM AUTHOR]

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$ , the real torus of dimension $2g$ , by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties. [ABSTRACT FROM AUTHOR]

Let |$p$| be an algebraic point of a projective variety |$X$| defined over a number field. Liouville inequality tells us that the norm at |$p$| of a non-vanishing integral global section of a hermitian line bundle over |$X$| is zero or it cannot be too small with respect to the |$\sup $| norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the moduli space of low degree rational curves on a low-degree complete intersection passing several suitable points is a complete intersection. The proof involves some relationships among some divisors on the Kontsevich spaces, advanced techniques of deformations of rational curves, and classical projective geometry. [ABSTRACT FROM AUTHOR]

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than 1 2 ⁢ (p - 1) {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map ƒ of P1 of degree d > 1 defined over K in terms of the multipliers of periodic points of f having the formally exact period n, with an explicit error estimate in terms of ƒ, n, and d. As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over K. Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish [label = −] a quantification of Silverman's and Ingram's recent comparison between the critical height and any ample height on the dynamical moduli space Md(Q) except for the flexible Lattès locus, an improvement of McMullen's finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and an explicit computation on the magnitude of the formally exact period of cycles, and a characterization of non-affine isotrivial rational maps defined over a function field C(X) of a complex normal projective variety X in terms of the growth of the degree of the multipliers of cycles. [ABSTRACT FROM AUTHOR]

The proposed work deals with a distinguished application of complex algebraic-geometric structure in nonlinear dynamical components of block ciphers. Due to the complex algebraic structure of the elliptic curves, their application in public-key cryptography has gained immense importance in recent years. The presented study introduces an effective application of elliptic curves in the nonlinear components, used in a symmetric-key cryptosystem. We propose that the group law defined on the rational points of an elliptic curve, over the binary field, offers extra-ordinary technical advantages, when used in the byte substitution process in block ciphers. For this purpose a specific elliptic curve over the Galois field F 2 4 is selected that is special as its elliptic group is of the same order as that of F 2 4 . This feature helps us defining a bijective map between the two structures that renders highly increased level of perplexity and confusion in data. The cryptographic forte of the proposed method is tested through most significant analyses and when compared with one of the most recent schemes, it shows outstanding results. [ABSTRACT FROM AUTHOR]

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