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Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch curves have already been studied from the parametric point of view, although numerical methods and approximations are often necessary for their computation. In this paper, implicit equations of Holditch curves of algebraic curves are studied. The implicit equations can be simply found from the computation of a resultant of two polynomials. With the same techniques, Holditch curves of two initial algebraic curves are also considered. Moreover, the use of implicit equations allows to find new and explicit parameterizations of non-trivial Holditch curves, such as in the case of having an ellipse as an initial curve. [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]

The ordinariness of elliptic curves is essential in proving various expected properties of elliptic curves in positive characteristic and can be extended to algebraic curves of arbitrary genus. The present paper deals with another kind of extension, i.e., ordinariness of dormant sl 2 -opers, or more generally, dormant sl n -opers. We prove that, for dormant sl 2 -opers on elliptic curves, this notion is essentially equivalent to the classical ordinariness. Moreover, the main result of the present paper asserts that the pull-back of an ordinary dormant sl n -oper on a general curve by a cyclic covering is ordinary whenever the order of its Galois group is prime to the characteristic of the base field. This result may be regarded as an analogue of a result by S. Nakajima for ordinary algebraic curves. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

In the present paper, we give an upper bound for the generic degree of the generalized Verschiebung between the moduli spaces of rank two stable bundles with trivial determinant. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. 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 Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Let $ \mathbb K$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $ \mathbb K$ , containing three points, $P_{1},P_{2}$ , and $P_{3}$ , such that $nP_{1}+P_{2}$ , $nP_{2}+P_{3}$ , and $nP_{3}+P_{1}$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_{1},P_{2},P_{3}\}$ , as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points $P_{1},P_{2}$ , and $P_{3}$. When $ \mathbb K=\overline { \mathbb F}_{q}$ , this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to give an effective version of the noncritical p -tame Belyĭ theorem. That is to say, we compute an explicit bound on the minimal degree of tamely ramified Belyĭ maps in positive characteristic which are unramified at a prescribed finite set of points. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale K -theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale K -theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen K -theory of if the Bass and Quillen–Lichtenbaum conjectures hold true for . [ABSTRACT FROM AUTHOR]

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in ℝ m with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field. [ABSTRACT FROM AUTHOR]

By taking full advantage of the structure of complex algebraic curves and by using compactness arguments, in this paper we give a self-contained proof that holomorphic algebraic functions verify a uniform Bernstein–Remez inequality. Namely, their growth over a bounded, open, complex set is uniformly controlled by their size on a compact complex subset of sufficiently high cardinality. Up to our knowledge, the first known demonstration on the existence of such an inequality for a specific subset of algebraic functions is contained in Nekhoroshev's 1973 breakthrough on the genericity of close-to-integrable Hamiltonian systems that are stable over long time. Despite its pivotal rôle, this passage of Nekhoroshev's proof has remained unnoticed so far. This work aims at extending and generalizing Nekhoroshev's arguments to a modern framework. We stress the fact that our proof is different from the one contained in Roytwarf and Yomdin's seminal work (1998), where Bernstein-type inequalities are proved for several classes of functions. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and uniqueness of limit cycles for socalled quadratic systems with a symmetrical solution: ... where (x, y) ∊ R², t ∊ R, aij, bij ∊ R, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic. [ABSTRACT FROM AUTHOR]

Let C be a stable curve of compact type such that two smooth curves with general moduli are bridged by a chain of n -elliptic curves, which will be called a TCBE curve. In the present paper, we investigate the existence of a smoothable limit linear series on C whose Brill-Noether number ρ satisfies − n ≤ ρ < 0. Through this work, we find some relationships between a family of TCBE curves of genus g and a Brill-Noether locus in the moduli space of stable curves of genus g. Further, these results show some relations among Brill-Noether loci. [ABSTRACT FROM AUTHOR]

In this paper we consider n-poised planar node sets, as well as more special ones, called G C sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ℓ is called k-node line for a node set $\mathcal X$ if it passes through exactly k nodes. An ( n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of $\mathcal X$ is used by each node in $\mathcal X\setminus M, $ meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C set $\mathcal {X}$ is used either by exactly $\binom {n}{2}$ nodes or by exactly $\binom {n-1}{2}$ nodes. We prove also similar statements concerning n-node or ( n − 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C sets. At the end we present a conjecture concerning any k-node line. [ABSTRACT FROM AUTHOR]

Let (E , V) be a general generated coherent system of type (n , d , n + m) on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of E to the semistability of the kernel of the evaluation map V ⊗ 𝒪 X → E. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler's Conjecture in some cases. The strongest results are obtained for type (2 , d , 4) , which is the first previously unknown case. [ABSTRACT FROM AUTHOR]

Abstract We study a generalization of genus-zero r -spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open r -spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the r th Gelfand–Dickey hierarchy. This parallels an analogous result for the open r -spin generating function in the companion paper Buryak et al. (2018) to this work. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. This method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations involving the given parametrization. As a consequence, we obtain an algorithm that constructs a parametrization which contains a given point, P , as a singularity as well as some additional information as for instance, the order of P , parameters corresponding to P , multiplicity of each parameter and the singularities in the first neighborhood of the singularity P. Highlights • We provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. • The method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations. • We obtain an algorithm that outputs a parametrization of a rational curve having singularities at some given input points. • In the algorithm, the singularity, P , the order of P , the parameters (and their multiplicity) corresponding to P , and the first neighborhood of P , are fixed. • We translate every detail of the definitions and resolutions into the language of parametric equations, which are quite helpful to CAGD. [ABSTRACT FROM AUTHOR]

For a Blaschke product B of degree d and λ on ∂ D , let ℓ λ be the set of lines joining each distinct two preimages in B − 1 ( λ ) . The envelope of the family of lines { ℓ λ } λ ∈ ∂ D is called the interior curve associated with B . In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let L λ be the set of lines tangent to ∂ D at the d preimages B − 1 ( λ ) and the trace of the intersection points of each two elements in L λ as λ ranges over the unit circle is called the exterior curve associated with B . In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this paper, for a Blaschke product of degree d , we give some geometrical properties that lie between the interior curve and the exterior curve. [ABSTRACT FROM AUTHOR]

• A new 7R unit is proposed to design pyramid and frustum mechanism. • A new RRR chain is proposed to fold the frustum into a bundle. • A general frame is proposed to design 1 DOF deployable mechanism. • Two methods are proposed to fold a polyhedron into a bundle. In this paper, a series of deployable polygon mechanisms (DGMs) and deployable polyhedron mechanisms (DPMs) are proposed. Firstly, we propose a new plane-symmetric 7R unit with folded lateral edges. Kinematics and suitable parameter values are calculated to realize full expansion. N-pyramid and N-frustum and a larger DPM are designed. Secondly, we propose a new RRR chain to connect existing deployable units. Threefold Bricard mechanism and alternative Bennett mechanism are used as two examples to fold triangular and square frustum into a bundle. Thirdly, we propose a general frame to design 1 DOF deployable mechanism based on the continuation method. Three improvements in accuracy, speed, and generality are made. Feasible points of general and special N-side DGMs are calculated (3≤N≤7) based on plane-symmetric hexagon 6R and heptagon 7R mechanisms. Our results not only unify most deployable units but also design many new ones and reconfigurable DGMs. Fourthly, we propose the polyhedron net method and the least joint angle method to fold a polyhedron into a bundle and reduce the number of links, joints, and DOF. Many DPMs are designed. [ABSTRACT FROM AUTHOR]

In this paper, a general theoretical study, from the perspective of the algebraic geometry, of the untrimmed bisector of two real algebraic plane curves is presented. The curves are considered in C 2 , and the real bisector is obtained by restriction to R 2 . If the implicit equations of the curves are given, the equation of the bisector is obtained by projection from a variety contained in C 7 , called the incidence variety, into C 2 . It is proved that all the components of the bisector have dimension 1. A similar method is used when the curves are given by parametrizations, but in this case, the incidence variety is in C 5 . In addition, a parametric representation of the bisector is introduced, as well as a method for its computation. Our parametric representation extends the representation in Farouki and Johnstone (1994b) to the case of rational curves. [ABSTRACT FROM AUTHOR]

In this paper, we will study Ciani curves in characteristic p ≥ 3 , in particular their standard forms C : x 4 + y 4 + z 4 + r x 2 y 2 + s y 2 z 2 + t z 2 x 2 = 0. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then r , s , t belong to F p 2 and C is maximal or minimal over F p 2 . Moreover, in this case we will provide a simple criterion in terms of r , s , t , p that tells whether C is maximal (resp. minimal) over F p 2 . [ABSTRACT FROM AUTHOR]

Generalized almost perfect nonlinear (GAPN) functions are a generalization of APN functions to finite fields of odd characteristic p introduced in 2017 by Kuroda and Tsujie. In this paper we deal with GAPN functions of monomial type. To this aim, we connect the GAPN property for a monomial function over F p n to the existence of suitable rational points of an algebraic curve defined over F p n . We give necessary conditions for a monomial function to be GAPN, providing the converse of recent results by Özbudak and Sălăgean and by Zha, Hu and Zhang. [ABSTRACT FROM AUTHOR]

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we aim for the theta function representation of quasi-periodic solution and related crucial quantities for a two-component generalization of Burgers' equation. Our tools include the theory of algebraic curves, meromorphic functions, Baker—Akhiezer functions and the Dubrovin-type equations for auxiliary divisor. Eith these tools, the explicit representations for above quantities are obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two. [ABSTRACT FROM AUTHOR]

Let f (X) ∈ F q r [ X ] be a q -polynomial. If the F q -subspace U = { (x q t , f (x)) | x ∈ F q n } defines a maximum scattered linear set, then we call f (X) a scattered polynomial of index t. The asymptotic behavior of scattered polynomials of index t is an interesting open problem. In this sense, exceptional scattered polynomials of index t are those for which U is a maximum scattered linear set in PG (1 , q m r) for infinitely many m. The classifications of exceptional scattered monic polynomials of index 0 (for q > 5) and of index 1 were obtained in [1]. In this paper we complete the classifications of exceptional scattered monic polynomials of index 0 for q ≤ 4. Also, some partial classifications are obtained for arbitrary t. As a consequence, the classification of exceptional scattered monic polynomials of index 2 is given. [ABSTRACT FROM AUTHOR]

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in F q n × n the idealizers have been proved to be isomorphic to finite fields of size at most q n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in F q n × n for n ≤ 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for n = 7 , q odd and for n = 8 , q ≡ 1 (mod 3). These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n ≥ 9. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus 4 over F 11. Our algorithm determining automorphism groups works for any nonhyperelliptic curve of genus 4 over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus 4 over F 11 obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory. [ABSTRACT FROM AUTHOR]