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In this paper, we study the algebraic limit cycle of degree 2 for the system x ˙ = - y + a 30 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = x + b 30 x 3 + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , which is called a linear type center plus cubic homogeneous polynomial differential system. We derive the normal form of this system with an algebraic limit cycle of degree 2. It is shown that, if an invariant algebraic curve of degree 2 is a limit cycle of the above mentioned system, then it is a unique limit cycle in the phase plane. [ABSTRACT FROM AUTHOR]

We prove in this paper that a hybrid van der Pol-Rayleigh oscillator x ˙ = y , y ˙ = − x − α y − β y 3 − x 2 y with (α , β) ∈ R 2 has no irreducible invariant algebraic curves in C [ x , y ] except for (α , β) = (0 , − 1) , or α 2 = − β − 1 / β + 2 , β ≠ 0 or α ≠ 0 , β = 1 , and that the oscillator has only one irreducible invariant algebraic curve in R [ x , y ] when (α , β) = (0 , − 1) , or α 2 = − β − 1 / β + 2 , β < 0 , or α ∈ R , β = 1. These together with that in Chen et al. (2021) [7] ensure that the hybrid van der Pol-Rayleigh oscillator has an algebraic limit cycle if and only if β = 1 and α < 0. This result solves the conjecture posed in that paper. [ABSTRACT FROM AUTHOR]

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins–Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5. [ABSTRACT FROM AUTHOR]

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters. [ABSTRACT FROM AUTHOR]

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras. [ABSTRACT FROM AUTHOR]

We prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev-Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions. [ABSTRACT FROM AUTHOR]

A number field K is primitive if K and Q are the only subextensions of K. Let C be a curve defined over Q . We call an algebraic point P ∈ C (Q ¯) primitive if the number field Q (P) is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d. For example, let C / Q be a hyperelliptic curve of genus g, and let 3 ≤ d ≤ g - 1 . Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group S d or A d . However, for any even d ≥ 4 , a hyperelliptic curve C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of S 2 ≀ S d / 2 . [ABSTRACT FROM AUTHOR]

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem L y = λ y , for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ , defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L − λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. [ABSTRACT FROM AUTHOR]

The Riccati polynomial differential systems are differential systems of the form x ′ = c 0 (x) , y ′ = b 0 (x) + b 1 (x) y + b 2 (x) y 2 , where c 0 and b i for i = 0 , 1 , 2 are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable y defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system. [ABSTRACT FROM AUTHOR]

We consider the following Kirchhoff problem - (a+b ∫R³ |▽u|²) Δu + (1+εV(x))u = |u|p-2u, where a,b > 0, and 2 < p < 6. Under suitable assumptions on V, by using the Lyapunov-Schmidt reduction method, we obtain the existence of multi-bump solutions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the upper bound and H¨older continuity for a general class of parametric elliptical variational-hemivariational inequalities via regularized gap functions. More precisely, we deliver a formulation of the elliptical variational-hemivariational inequalities in the case of the perturbed parameters governed by both the set of constraints and the mappings (for brevity, PEVHI (CM)). Based on the arguments of monotonicity and properties of the Clarke's generalized directional derivative, we establish an upper bound result for the PEVHI (CM) and provide the H¨older continuity of the solution mapping for the PEVHI (CM) under suitable assumptions on the data. [ABSTRACT FROM AUTHOR]

In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider Stäckel transformbetween two pairs of Stäckel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between Stäckel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems. [ABSTRACT FROM AUTHOR]

Propagation is a standard way of producing certain new conformal blocks from old ones that corresponds to the geometric procedure of adding new distinct points to a pointed compact Riemann surface. On the other hand, sewing conformal blocks corresponds to sewing compact Riemann surfaces. In this article, we clarify the relationships between these two procedures. Most importantly, we show that, "sewing and propagation are commuting procedures." More precisely: letϕbe a conformal block associated to a vertex operator algebra 𝕍 and a compact Riemann surface to be sewn, and let ≀𝑛ϕ be its 𝑛-times propagation. If the sewing ˜ 𝒮ϕ converges, then ˜ 𝒮 ≀𝑛 ϕ (the sewing of ≀𝑛ϕ) automatically converges, and it equals ≀𝑛 ˜ 𝒮ϕ (the 𝑛-times propagation of the sewing ˜ 𝒮ϕ). The proof of this result relies on establishing the propagation of conformal blocks associated to holomorphic families of compact Riemann surfaces. We prove this in our paper using the idea that, "propagation is itself a sewing followed by an analytic continuation." This result generalizes previous ones on single Riemann surfaces [Zhu94, FB04], and supplements those on algebraic families of complex algebraic curves [Cod19, DGT19a]. The results in this paper will be used in [Gui21] as the main technical tools to relate the (genus-0) permutation-twisted V⊗k-conformal blocks (i.e. intertwining operators) and the untwisted V-conformal blocks (of possibly higher genera). [ABSTRACT FROM AUTHOR]

In this work, the existence of a polynomial system of degree four with an invariant triangle containing at least four small-amplitude limit cycles is proved. This result improves the result obtained in[2]. [ABSTRACT FROM AUTHOR]

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The present paper is devoted to a scheme-theoretic analog of the Fredholm theory. The continuity of the index function over the coordinate ring of an algebraic variety is investigated. It turns out that the index is closely related to the filtered topology given by finite products of maximal ideals. We prove that a variety over a field possesses the index function on nonzero elements of its coordinate ring iff it is an algebraic curve. In this case, the index is obtained by means of the multiplicity function from its normalization if the ground field is algebraically closed. [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [ABSTRACT FROM AUTHOR]

Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A(x) - A(y) = 0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A0d, d ≤1, into compositions of rational functions can be obtained from decompositions of a single iterate A0N for N large enough. [ABSTRACT FROM AUTHOR]

The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties. [ABSTRACT FROM AUTHOR]

We provide constructions of 3 classical curves, using novel approaches, based on tools for automated exploration and reasoning, especially for the determination of geometric loci. Dragging and animations are the core features in use. The different ways yield an output based either on numerical computations or on symbolic computations (these use Gröbner bases packages). These curves, which are generally presented as separate cases, appear here in a unifying frame. This work is a contribution to the study of plane curves and to a working frame aimed at developing the dialog between different kinds of mathematical software. [ABSTRACT FROM AUTHOR]

We give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperkähler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold not containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we obtain the identification of the period point of the LLSvS eightfold with that of the Fano variety. We discuss the derived Torelli theorem for cubic fourfolds. [ABSTRACT FROM AUTHOR]

Let F q 2 be the finite field with q 2 elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over F q 2 with many rational points. The curves constructed are Kummer covers or fiber products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves. [ABSTRACT FROM AUTHOR]

It is challenging to characterize the intrinsic geometry of high-degree algebraic curves with lower-degree algebraic curves. The reduction in the curve’s degree implies lower computation costs, which is crucial for various practical computer vision systems. In this paper, we develop a characteristic mapping (CM) to recursively degenerate $\mathbf {3n}$ points on a planar curve of $n$ th order to $\mathbf {3(n-1)}$ points on a curve of $\mathbf {(n-1)}$ th order. The proposed characteristic mapping enables curve grouping on a line, a curve of the lowest order, that preserves the intrinsic geometric properties of a higher-order curve (ellipse). We prove a necessary condition and derive an efficient arc grouping module that finds valid elliptical arc segments by determining whether the mapped three points are colinear, invoking minimal computation. We embed the module into two latest arc-based ellipse detection methods, which reduces their running time by 25% and 50% on average over five widely used data sets. This yields faster detection than the state-of-the-art algorithms while keeping their precision comparable or even higher. Two CM embedded methods also significantly surpass a deep learning method on all evaluation metrics. [ABSTRACT FROM AUTHOR]

The minimization of an energy functional is the main ingredient of several segmentation and geometric modeling problems. When the solution of this kind of optimization problem is described by a curve, the most popular approach consists in representing the curve as a parametric curve and to compute the minimum in terms of the free parameters of the curve. In free form design tasks, the fairness (energy) functional depends of the arc length and the bending energy of the curve and the classical approach requires to compute first and second derivatives. This work presents a Hermite interpolating subdivision scheme, based on Bézier rational curves, with local tension parameters and discusses an efficient software implementation of the algorithm for energy minimization of the functional. The curve that minimizes the functional is called the fair curve, and it shows excellent properties to be used for design purposes. The novelty of the proposed method lies in the fact it is derivative free. Also we include a discussion of the implementation of our method and show some numerical results. [ABSTRACT FROM AUTHOR]

Francis Sowerby Macaulay began his career working on Brill and Noether's theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay's work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected. [ABSTRACT FROM AUTHOR]

Copyright of Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales is the property of Academia Colombiana de Ciencias Exactas, Fisicas y Naturales and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper is an introduction to the use of the classical Newton–Puiseux procedure, oriented towards an algorithmic description of it. This procedure allows to obtain polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and is almost self-contained. We illustrate the use of the algorithm first in a completely worked out example of a curve with a point of multiplicity 6, and secondly, in the study of triple points on reduced plane curves. [ABSTRACT FROM AUTHOR]

In analytic geometry, Bézout's theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in R 2 and A K 2 under fold point, where char K = 0. First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in R 2 . Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in A K 2 are equivalent and then give the exact expression of the intersection multiplicity of two curves at some point in A K 2 under fold point. [ABSTRACT FROM AUTHOR]

In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, García Barroso and Płoski studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups. They described such semigroups with the maximum conductor. In this paper we prove that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. Our proof is constructive, explicitly describing families that achieve a given value as its conductor. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study and define some algebraic curves with neutrosophic variables in neutrosophic real field R(I), where we study what are the relationships between classical algebraic curves and neutrosophic algebraic curves depending on the geometric isometry (AH-Isometry). [ABSTRACT FROM AUTHOR]

Among several methods employed for sizing thickeners available in the literature, the Kynch, Biscaia Jr., Talmadge and Fitch, Roberts, Coe and Clevenger, and Oltmann methods use experimental data from sedimentation curves and graphical approaches. By using the Weibull distribution, it is possible to represent sedimentation curves with algebraic equations, which does not require the use of graphical approaches and provides more accuracy and speed for sizing calculations. In the present work, the main objective is the development of a set of equations for sizing continuous thickeners, for six conventional methods found in the literature, using the Weibull model. A comparative analysis of calculated and literature diameters for each graphical method presented variations between 0.73% and 8.93%. The use of the Weibull model presented the best accuracy for the Biscaia Jr. method, with a 0.73% average absolute error. [ABSTRACT FROM AUTHOR]

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k > θ 1 > θ 2 > θ 3 , and that the triple (θ 1 , θ 2 , θ 3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7 , there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ≥ 9 . [ABSTRACT FROM AUTHOR]

Using the three-term Weierstrass relation for theta functions and the properties of Hecke-type double sums and Appell-Lerch sums, we give new and simple proofs for the seventh order mock theta conjectures. [ABSTRACT FROM AUTHOR]

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e , respectively, with d e = n , and with d + e − 1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists. [ABSTRACT FROM AUTHOR]

We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaker notion of generic position. As distinct to previous work, our sweep generic position does not require that x -critical points have different x -coordinates. The complexity of achieving this sweep generic position property is thus no longer a bottleneck in term of complexity. The bit complexity of our algorithm is O ˜ (d 18 + d 17 τ) where d and τ bound the degree and the bitsize of the integer coefficients, respectively, of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of d 2. [ABSTRACT FROM AUTHOR]

Gauss's class number one problem, solved by Heegner, Baker, and Stark, asked for all imaginary quadratic fields for which the ideal class group is trivial. An application of this solution gives all elliptic curves that can be defined over the rationals and have a large endomorphism ring (CM). Analogously, to get all CM curves of genus two defined over the smallest number fields, we need to find all quartic CM fields for which the CM class group (a quotient of the ideal class group) is trivial. We solve this CM class number one problem. We prove that the list given in Bouyer–Streng [LMS J Comput Math 18(1):507–538, 2015, Tables 1a, 1b, 2b, and 2c] of maximal CM curves of genus two defined over the reflex field is complete. We also prove that there are exactly 21 simple CM curves of genus two over C that can be defined over Q . [ABSTRACT FROM AUTHOR]

We consider the Rayleigh equation x + λ(x2/3 - 1)x˙ + x = 0 depending on the real parameter λ and construct a Poincaré-Bendixson annulus Aλ in the phase plane containing the unique limit cycle Γλ of the Rayleigh equation for all λ > 0. The novelty of this annulus consists in the fact that its boundaries are algebraic curves depending on λ. The polynomial defining the interior boundary represents a special Dulac-Cherkas function for the Rayleigh equation which immediately implies that the Rayleigh equation has at most one limit cycle. The outer boundary is the diffeomorphic image of the corresponding boundary for the van der Pol equation. Additionally we present some equations which are linearly topologically equivalent to the Rayleigh equation and provide also for these equations global algebraic Poincaré-Bendixson annuli. [ABSTRACT FROM AUTHOR]

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic. [ABSTRACT FROM AUTHOR]

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve. [ABSTRACT FROM AUTHOR]

Let C be an algebraic curve and c be an analytically irreducible singular point of C. The set ℒ∞(C)𝑐 of arcs with origin c is an irreducible closed subset of the space of arcs on C. We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ℒ∞(C)𝑐 the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space. [ABSTRACT FROM AUTHOR]

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C , implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them. [ABSTRACT FROM AUTHOR]

This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points. [ABSTRACT FROM AUTHOR]

We consider tropical versions of Hassett's moduli spaces of weighted stable curves Mtropg,A, SMtropg,A and Δg,A associated to a weight datum A... their associated graph complexes G(g,A), and study the topology of these spaces as A changes. We show that for fixed g and n, there are particular filtrations of these topological spaces and their graph complexes which may be used to compute the reduced rational homology of Δg,A and the top weight cohomology of the moduli space Mg,A of smooth (g,A)-stable algebraic curves. [ABSTRACT FROM AUTHOR]

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The hull of a linear code C is the intersection of C with its dual C ⊥ . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]