Search

We construct qualitatively new examples of superabundant tropical curves which are non‐realisable in genuses 3 and 4. These curves are in R3${\mathbb {R}}^3$ and R4${\mathbb {R}}^4$, respectively, and have properties resembling canonical embeddings of genus 3 and 4 algebraic curves. In particular, the genus 3 example is a degree 4 planar tropical curve, and the genus 4 example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to 1. Non‐realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves. [ABSTRACT FROM AUTHOR]

We prove that any geometrically connected curve X$X$ over a field k$k$ is an algebraic K(π,1)$K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of Z/nZ$\mathbb {Z}/n\mathbb {Z}$‐modules, with n$n$ invertible in k$k$, is canonically isomorphic to the cohomology of its corresponding π1(X)$\pi _1(X)$‐module. To this end, we explicitly construct some Galois coverings of X$X$ corresponding to Galois coverings of the normalisation of its irreducible components. When k$k$ is finite or algebraically closed, we precisely describe finite quotients of π1(X)$\pi _1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products H1×H1→H2$\operatorname{H}^1\times \operatorname{H}^1\rightarrow \operatorname{H}^2$ and H1×H2→H3$\operatorname{H}^1\times \operatorname{H}^2\rightarrow \operatorname{H}^3$ in étale cohomology in terms of finite group cohomology. [ABSTRACT FROM AUTHOR]

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to k$k$ times the canonical divisor for k∈Z⩾1$k \in \mathbb {Z}_{\geqslant 1}$. In this article, we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non‐Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri‐canonical divisor. To do so, we introduce tropical normalized covers as special instances of cyclic tropical Hurwitz covers and reduce the realizability problem for pluri‐canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of Möller–Ulirsch–Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of k$k$‐differentials in the work of Bainbridge–Chen–Gendron–Grushevsky–Möller. [ABSTRACT FROM AUTHOR]

The control problem of the rotary double inverted pendulum (double Furuta pendulum) is nontrivial because of underactuation and strong nonlinearities in the associated state‐space model. The system has three degrees of freedom (one actuated and two unactuated joints) while receiving only one control input. In this article, a novel nonlinear optimal (H‐infinity) control approach is developed for the dynamic model of the rotary double inverted pendulum. First, the dynamic model of the double pendulum undergoes approximate linearization with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a temporary operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. At a next stage a stabilizing H‐infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation has to be solved at each time‐step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. To implement state estimation‐based control without the need to measure the entire state vector of the rotary double‐pendulum the H‐infinity Kalman filter is used as a robust state observer. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the rotary double inverted pendulum under moderate variations of the control input. [ABSTRACT FROM AUTHOR]

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]

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 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]

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 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 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]

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by‐product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time‐dependent exponential factor. [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]

Consider a field k${\mathsf {k}}$ of characteristic 0$\hskip.001pt 0$, not necessarily algebraically closed, and a fixed algebraic curve f=0$f=0$ defined by a tame polynomial f∈k[x,y]$f\in {\mathsf {k}}[x,y]$ with only quasi‐homogeneous singularities. We prove that the space of holomorphic foliations in the plane Ak2$\mathbb {A}^2_{\mathsf {k}}$ having f=0$f=0$ as a fixed invariant curve is generated as k[x,y]${\mathsf {k}}[x,y]$‐module by at most four elements, three of them are the trivial foliations fdx,fdy$fdx,fdy$ and df$df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen–Suslin theorem, we show that for a suitable field extension K${\mathsf {K}}$ of k${\mathsf {k}}$ such a module over K[x,y]${\mathsf {K}}[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well‐known examples, K=k${\mathsf {K}}={\mathsf {k}}$. [ABSTRACT FROM AUTHOR]

We show that U(k)$U(k)$‐invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in gl(k,C)${\mathfrak {g} \mathfrak {l}}(k,{\mathbb {C}})$ correspond to algebraic curves C of genus (k−1)2$(k-1)^2$, equipped with a flat projection π:C→P1$\pi :C\rightarrow {\mathbb {P}}^1$ of degree k, and an antiholomorphic involution σ:C→C$\sigma :C\rightarrow C$ covering the antipodal map on P1${\mathbb {P}}^1$. [ABSTRACT FROM AUTHOR]

A Kodaira fibration is a non‐isotrivial fibration f:S→B$f: S\rightarrow B$ from a smooth algebraic surface S to a smooth algebraic curve B such that all fibers are smooth algebraic curves of genus g. Such fibrations arise as complete curves inside the moduli space Mg$\mathcal {M}_g$ of genus g algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case g=3$g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of M3$\mathcal {M}_3$ parametrizing curves whose Jacobians have extra endomorphisms. [ABSTRACT FROM AUTHOR]

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable non‐planar curves of genus 0 and degree d$d$ in P3∖P1${\mathbb {P}}^3\backslash {\mathbb {P}}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non‐reductive Lie group. [ABSTRACT FROM AUTHOR]

Let C$C$ be an algebraic curve of genus g$g$ and L$L$ a line bundle over C$C$. Let MSC(n,L)$\mathcal {M}S_C(n,L)$ and MOC(n,L)$\mathcal {M}O_C(n,L)$ be the moduli spaces of L$L$‐valued symplectic and orthogonal bundles, respectively, over C$C$ of rank n$n$. We construct rational curves of Hecke type on these moduli spaces which generalize the Hecke curves on the moduli space of vector bundles. As a main result, we show that these curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its by‐products, we show the non‐abelian Torelli theorem and compute the automorphism group of the moduli spaces. [ABSTRACT FROM AUTHOR]

The supersingular isogeny Diffie–Hellman key exchange protocol (SIDH) was introduced by Jao and De Feo in 2011. SIDH operates on supersingular elliptic curves defined over Fp2, where p is a large prime number of the form p=4eA3eB−1 and eA and eB are positive integers such that 4eA≈3eB. A variant of the SIDH protocol, dubbed extended SIDH (eSIDH), is presented. The eSIDH makes use of primes of the form p=4eAℓBeBℓCeCf−1. Here ℓB and ℓC are two small prime numbers; f is a cofactor; and eA, eB, and eC are positive integers such that 4eA≈ℓBeBℓCeC. It is shown that for many relevant instantiations of the SIDH protocol, this new family of primes enjoys faster field arithmetic than the one associated with traditional SIDH primes. Furthermore, its richer opportunities for parallelism yield a noticeable speed‐up factor when implemented on multicore platforms. A supersingular isogeny key encapsulation (SIKE) instantiation using the prime eSIDH‐p765 yields an acceleration factor of 1.06, 1.15 and 1.14 over a SIKE instantiation with the prime SIKE‐p757 when implemented on k = {1, 2, 3}‐core processors. To the authors' knowledge, this work reports the first multicore implementation of SIDH and SIKE. [ABSTRACT FROM AUTHOR]

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity. [ABSTRACT FROM AUTHOR]

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work, we extend this methodology to a hierarchy of nonclassical orthogonal polynomials on disk slices and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (nonclassical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and biharmonic equations. In this paper, we focus on constant Dirichlet boundary conditions, as well as zero Dirichlet and Neumann boundary conditions, with other types of boundary conditions requiring future work. [ABSTRACT FROM AUTHOR]

The amoeba of a complex curve in the 2‐dimensional complex torus is its image under the projection onto the real parts of the logarithmic coordinates. A toric cycle in the complement to a curve is a fiber of this projection over a point in the complement to the amoeba of the curve. We consider amoebas of complex algebraic curves defined by so‐called Harnack polynomials. We prove that toric cycles are homologically independent in the complement to a such curve. [ABSTRACT FROM AUTHOR]

The problem of building a smooth surface containing given points or curves is actual due to development of industry and computer technology. Previously used for those purposes, shells of zero Gaussian curvature and minimal surfaces based on isotropic analytic curves are restricted in their consumer properties. To expand the possibilities regarding the shaping of surfaces we propose the method of constructing surfaces based on isotropic fractional-rational curves. The surfaces are built using flat isothermal and orthogonal grids and on the basis of the Weierstrass method. In the latter case, the surfaces are minimal. Examples of surfaces that were built according to the proposed method are given. [ABSTRACT FROM AUTHOR]

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree. [ABSTRACT FROM AUTHOR]

Rapoport–Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their l‐adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the l‐adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport–Zink spaces which are Hodge type and Hodge–Newton reducible. The main strategy is to embed such a Rapoport–Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan. [ABSTRACT FROM AUTHOR]

In this article we construct for any prime power q and odd n⩾5, a new Fq2n‐maximal curve Xn. Like the Garcia–Güneri–Stichtenoth maximal curves, our curves generalize the Giulietti–Korchmáros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q2−1)(qn+1) automorphisms. Further, we show that unless q=2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of Fq2n‐maximal curves, by considering some Galois subcovers of Xn. [ABSTRACT FROM AUTHOR]

In this paper, a class of switching systems which have an invariant conic x2+cy2=1,c∈R, is investigated. Half attracting invariant conic x2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved. [ABSTRACT FROM AUTHOR]

We construct the first non‐trivial examples of complete families of non‐degenerate smooth space curves, and show that the base of such a family cannot be a rational curve. Both results rely on the study of the strong semistability of certain vector bundles. [ABSTRACT FROM AUTHOR]

Abstract: Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature ⩾ κ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the MCP sense. The counterexample is given by the Grushin half‐plane, which satisfies the MCP ( 0 , N ) if and only if N ⩾ 4, while its double satisfies the MCP ( 0 , N ) if and only if N ⩾ 5. [ABSTRACT FROM AUTHOR]

Abstract: In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely many Lucas–Carmichael numbers (that is, squarefree, composite integers n such that for every prime p that divides n, p + 1 | n + 1) . [ABSTRACT FROM AUTHOR]

In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariant σ. Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant σ of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone ONCv0⁎. Finally, some relevant counterexamples are indicated. [ABSTRACT FROM AUTHOR]

Abstract: We consider all one‐parameter families of smooth curves degenerating to a singular curve X and describe limits of linear series along such families. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. We introduce the notion of level‐δ limit linear series on X to describe these limits, where δ is the singularity degree of the total space of the degeneration at P. If the total space is regular, that is, δ = 1, we recover the limit linear series introduced by Osserman in . So we extend his treatment to a more general setup. In particular, we construct a projective moduli space G d , δ r ( X ) parameterizing level‐δ limit linear series of rank r and degree d on X, and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize by associating with each exact level‐δ limit linear series g on X a closed subscheme P ( g ) ⊆ X ( d ) of the dth symmetric product of X, and showing that, if g is a limit of linear series on the smooth curves degenerating to X, then P ( g ) is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X. [ABSTRACT FROM AUTHOR]

Abstract: The quintic family must be the most studied family of Calabi‐Yau threefolds. Particularly symmetric members of this family are known to admit quotients by freely acting symmetries isomorphic to Z 5 × Z 5. The corresponding quotient manifolds may themselves be symmetric. That is, they may admit symmetries that descend from the symmetries that the manifold enjoys before the quotient is taken. The formalism for identifying these symmetries was given a long time ago by Witten and instances of these symmetric quotients were given also, for the family P 7 [ 2 , 2 , 2 , 2 ], by Goodman and Witten. We rework this calculation here, with the benefit of computer assistance, and provide a complete classification. Our motivation is largely to develop methods that apply also to the analysis of quotients of other CICY manifolds, whose symmetries have been classified recently. For the Z 5 × Z 5 quotients of the quintic family, our list contains families of smooth manifolds with symmetry Z 4, Dic 3 and Dic 5, families of singular manifolds with four conifold points, with symmetry Z 6 and Q 8, and rigid manifolds, each with at least a curve of singularities, and symmetry Z 10. We intend to return to the computation of the symmetries of the quotients of other CICYs elsewhere. [ABSTRACT FROM AUTHOR]

By introducing a trigonal curve [ABSTRACT FROM AUTHOR]

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K, then there is a natural bijection between the set of equivalence classes of elliptic subcovers of [ABSTRACT FROM AUTHOR]

Let X be a measurable space, let [ABSTRACT FROM AUTHOR]

Using the notion of generalized divisors introduced by Hartshorne (see []), we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli-type theorem for line bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve. [ABSTRACT FROM AUTHOR]

In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub-super solution method, fixed point theorem, Leray-Schauder degree theory and comparison theorems. [ABSTRACT FROM AUTHOR]

We compute the class of the classifying stack of the special orthogonal group in the Grothendieck ring of stacks, and check that it is equal to the multiplicative inverse of the class of the group. [ABSTRACT FROM AUTHOR]

We apply Floquet theory of periodic coefficient second-order ODEs to an elastic waveguide. The waveguide is modeled as a uniform elastic string periodically supported by a discontinuous Winkler elastic foundation and, as a result, a Hill equation is found. The fundamental solutions, the stability regions, and the dispersion curves are determined and then plotted. An asymptotic approximation to the dispersion curve is also given. It is further shown that the end points of the band gap structure correspond to periodic and semiperiodic solutions of the Hill equation. [ABSTRACT FROM AUTHOR]

We prove that the (logarithmic) Mahler measure m(P) of P(x, y) = x + 1/x + y + 1/y + 3 is equal to the L-value 2L'(E, 0) attached to the elliptic curve E : P(x, y) = 0 of conductor 21. In order to do this, we investigate the measure of a more general Laurent polynomial Pa,b,c(x, y) = a(x + 1/x)+ b(y + 1/y)+ c and show that the wanted quantity m(P) is related to a 'half-Mahler' measure of Pữ(x, y) = P√7,1,3(x, y). In the finale, we use the modular parametrization of the elliptic curve Pữ(x, y) = 0, again of conductor 21, due to Ramanujan and the Mellit-Brunault formula for the regulator of modular units. [ABSTRACT FROM AUTHOR]

With the aid of Lenard recursion equations, we derive the Wadati-Konno-Ichikawa hierarchy. Based on the Lax matrix, an algebraic curve of arithmetic genus n is introduced, from which Dubrovin-type equations and meromorphic function φ are established. The explicit theta function representations of solutions for the entire WKI hierarchy are given according to asymptotic properties of φ and the algebro-geometric characters of . Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We formulate a conjecture about the distribution of the canonical height of the lowest non-torsion rational point on a quadratic twist of a given elliptic curve, as the twist varies. This conjecture seems to be very deep and we can prove only partial results in this direction. [ABSTRACT FROM AUTHOR]

There have been recent advances in the construction of algebraic curves for certain classes of string solutions in the context of the AdS/CFT correspondence. In this paper we obtain the Lax operators and associated spectral curves for circular and folded string solutions in AdS [ABSTRACT FROM AUTHOR]

We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base field. As an application, we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field is, in a suitable sense, the branch of an algebraic curve. [ABSTRACT FROM AUTHOR]

We investigate [ABSTRACT FROM AUTHOR]

We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number. [ABSTRACT FROM AUTHOR]

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED : y² + Dy = x³ (D ϵ ℚ*) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L : ℚ] often becomes small. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that [ABSTRACT FROM AUTHOR]

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP-complete. We also construct an infinite family [ABSTRACT FROM AUTHOR]