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For a smooth projective curve, we derive a closed formula for the generating series of its Gromov–Witten invariants in genus g$g$ and degree zero. It is known that the calculation of these invariants can be reduced to that of the λg$\lambda _g$ and λg−1$\lambda _{g-1}$ integrals on the moduli space of stable algebraic curves. The closed formula for the λg$\lambda _g$ integrals is given by the λg$\lambda _g$ conjecture, proved by Faber and Pandharipande. We compute in this paper the λg−1$\lambda _{g-1}$ integrals via solving the degree zero limit of the loop equation associated to the complex projective line. [ABSTRACT FROM AUTHOR]

The paper explains the meaning of the terms osculation circle of a conic and evolute of a conic in the Euclidean plane. One of the ways to derive their equations is also given. [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]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [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.)

Given a smooth, projective curve Y, a finite group G and a positive integer n we study smooth, proper families X → Y x S → S of Galois covers of Y with Galois group isomorphic to G branched in n points, parameterized by algebraic varieties S. When G is with trivial center we prove that the Hurwitz space HnG (Y) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary G we prove that HnG (Y) is a coarse moduli variety. For families of pointed Galois covers of (Y, y0) we prove that the Hurwitz space HnG (Y, y0) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group G. We use classical tools of algebraic topology and of complex algebraic geometry. [ABSTRACT FROM AUTHOR]

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered. [ABSTRACT FROM AUTHOR]

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyper-elliptic type. These Miura maps induce Miura maps between Stäckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for Stäckel systems. [ABSTRACT FROM AUTHOR]

In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If g ≥ 4 , it is not even known whether there exists such a curve of genus g in general characteristic p > 0 , and in the case of g = 4 , several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity O ~ (p 4) in theory but O ~ (p 3) in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for p up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every p with p ≡ 2 (mod 3) . [ABSTRACT FROM AUTHOR]

In analogy with the Manin–Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel–Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus |${g\geq 2}$|⁠ , which are biconnected and have edge lengths that are "sufficiently irrational" in a precise sense. Under these assumptions, the number of torsion points is bounded by |$3g-3$|⁠. Next, we study bounds on the number of torsion points in the image of higher-degree Abel–Jacobi embeddings, which send |$d$| -tuples of points to the Jacobian. This motivates the definition of the "independent girth" of a graph, a number that is a sharp upper bound for |$d$| such that the higher-degree Manin–Mumford property holds. [ABSTRACT FROM AUTHOR]

Let X be a curve defined over ℚ and let t ∈ ℚ (X) be a non-constant rational function on X of degree v ≥ 2. For every rational number a / b pick a point P a / b ∈ X (ℚ ¯) such that t (P a / b) = a / b. In this paper, we obtain lower bounds on the number of distinct fields among ℚ (P a / b) with 1 ≤ a , b ≤ N under some assumptions on t. We show that if t has a pole of order at least 2 or if there is a rational number α such that t − α has a zero of order at least 2, then the set { ℚ (P a / b) | 1 ≤ a , b ≤ N } contains ≫ N 2 (log N) 2 elements. We also obtain partial results when t does not have a pole of order at least two. [ABSTRACT FROM AUTHOR]

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]

For a long time, it was believed that Alfred Kempe proved (1876) a theorem about the possibility of plotting an arbitrary planar algebraic curve by parts using hinge mechanisms (linkages). However, at the end of the 20th century, specialists in algebraic geometry, having rediscovered and developed this result in modern language, began to assert that Kempe's reasoning contained significant gaps and errors. In the author's opinion, these claims are unfounded. In this paper, by a detailed analysis of the notions of a linkage and its configuration space, the author tries to substantiate this point of view. [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]

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field F q of characteristic p reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the possible intersection multiplicities of its strict dual curve with the linear system of hyperplanes. Among other consequence, using a result by Homma, we are able to construct nonreflexive space curves such that their tangent surfaces are nonreflexive as well, and the image of a generic point by a Frobenius map is in its osculating hyperplane. We also obtain generalizations and improvements of some known results of the literature. [ABSTRACT FROM AUTHOR]

We study the spaces of twisted conformal blocks attached to a $\Gamma$ -curve $\Sigma$ with marked $\Gamma$ -orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$ , where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$ , then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$ -curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$ -invariance of Weil restriction associated to $\Sigma$ and the simply connected simple algebraic group $G$ with Lie algebra $\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr {G}$ -torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport). [ABSTRACT FROM AUTHOR]

Given a finite set of points S ⊂ R d , a k-set of S is a subset A ⊂ S of size k which can be strictly separated from S \ A by a hyperplane. Similarly, a k-facet of a point set S in general position is a subset Δ ⊂ S of size d such that the hyperplane spanned by Δ has k points from S on one side. For a probability distribution P on R d , we study E P (k , n) , the expected number of k-facets of a sample of n random points from P. When P is a distribution on R 2 such that the measure of every line is 0, we show that E P (k , n) = O (n (k + 1) 1 / 4) . Our argument is based on a technique by Bárány and Steiger. We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the k-edge graph of a set of points. We also study a variation on the k-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by Bárány and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of k-sets which are tight up to logarithmic factors. [ABSTRACT FROM AUTHOR]

We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. [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]

We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves. [ABSTRACT FROM AUTHOR]

In the paper Targino (Outer Lipschitz Geometry of Complex Algebraic Plane Curves. International Mathematics Research Notices, rnac202. https://doi.org/10.1093/imrn/rnac202, 2022) Renato Targino shows that bi-Lipschitz type of a plane curve is determined by the local ambient topological properties of curves. Here we show that it is no longer true in higher dimensions. However we show that the bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane. [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]

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting. [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]

Over the complex numbers, Plücker's formula computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in A1$\mathbb {A}^1$‐homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field. [ABSTRACT FROM AUTHOR]

We introduce a new generalization of -congruent numbers by defining a rational -parallelogram envelope. It turns out that the existence of a rational -parallelogram envelope associated to an n is connected to the property of n being a -congruent number. Then, we study some problems related to the rational -parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of Ochiai, where only the case = π / 2 was considered. Moreover, we answer open questions raised by Ochiai in the more general setting of an arbitrary Pythagorean angle. [ABSTRACT FROM AUTHOR]

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, C , implicitly or parametrically defined. The asymptotes of a curve C reflect the status of C at points with sufficiently large coordinates. It is well known that an asymptote of a curve C is a line such that the distance between C and the line approaches zero as they tend to infinity. However, a curve C may have more general curves than lines describing the status of C at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field. [ABSTRACT FROM AUTHOR]

We study cohomologies of a curve with an action of a finite p-group over a field of characteristic p. Assuming the existence of a certain "magical element" in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p-group cover has a "magical element". As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order. [ABSTRACT FROM AUTHOR]

The substitution box (S-box) is the only nonlinear component of any block cipher. It hides the relationship between the key and the cipher text. Several methods are available in the literature for creating S-boxes of different sizes. However, the need to design 4 × 4 S-boxes for lightweight encryption is in demand. The suggested method first generates the initial S-boxes from the elliptic curve points, and then the S-boxes are evolved with the help of algebraic group structures. Algebraic group structures have the potential to improve their cryptographic strength. The cryptographic properties of the suggested S-boxes, such as nonlinearity, algebraic degree, branch number, and bijectivity, are analyzed. We also present a comparative analysis of the suggested S-boxes with different standard S-boxes. The offered S-boxes are utilized in an image steganography scheme. First, the secret image is encrypted through the combination of the substitution operation and the bitwise XOR of the key. The enciphered image is hidden in the cover image using the least significant bit of steganographic scheme. Experimental results show that the suggested algorithm has a high security level and better stego-image quality. [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]

We present a complete classification of complex plane algebraic curves equipped with the outer metric up to bi-Lipschitz homeomorphism. [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]

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form ym=ϕ(x)$y^{m} = \phi (x)$ in R2${\mathbb {R}}^2$ where m=1,2$m = 1, 2$ and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree 0,...,N$0, \ldots , N$ are computed via the Lanczos algorithm in O(Nd4)$\mathcal {O}(Nd^4)$ operations. [ABSTRACT FROM AUTHOR]

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in X (F ¯ q) \ X (F q 6) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X (F q 7) \ X (F q) and we give a bound on the Feng–Rao minimum distance d ORD . For q = 3 we provide a table that also reports the exact values of d ORD . As a further application we construct quantum codes from F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

In this paper we prove that the planar self-assembling micelle system dx/dt = μ − xy² − x(r + α), dy/dt = rx + xy² − ηy, has no Liouvillian, polynomial and Darboux first integrals. Moreover, we show that the system has only one irreducible Darboux polynomial ax + zy − μ with the cofactor being − a if and only if η = a via the weight homogeneous polynomials and only two irreducible exponential factors ex+y and e<(x+y)² with cofactors μ − zx − ny and 2(x + y)(μ − ax − ηy) respectively with (x + y − μ/a )λ e−aλt, be the unique Darbox invariant of system. [ABSTRACT FROM AUTHOR]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [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.)

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and when possible describe the structure as cone over an explicitly given variety. We determine the dimension of the moduli space for all semigroups of genus seven. [ABSTRACT FROM AUTHOR]