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Let X \subset \mathbb {C}^{2n} be an n-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of X. Moreover, we compute its singularities as well as degree, genus and Euler characteristic for X_d being a generic (smooth and transversal to the line at infinity) curve of degree d in \mathbb {C}^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]

We study one variable meromorphic functions mapping a planar real algebraic set A to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain A, these meromorphic functions must be rational. In particular, when A is the standard unit circle, we obtain a one dimensional analog of Poincaré [Acta Math. 2 (1883), pp. 97–113], Tanaka [J. Math. Soc. Japan 14 (1962), pp. 397–429] and Alexander's [Math. Ann. 209 (1974), pp. 249–256] rationality results for 2m-1 dimensional sphere in \mathbb {C}^m when m\ge 2. [ABSTRACT FROM AUTHOR]

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields \mathbb {Q}(\sqrt {t^4-n^2}) as t varies over positive integers for a congruent number n. Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form \mathbb {Q}(\sqrt {n^2k^4-1}) for positive integers k and congruent numbers n whose elliptic curves have algebraic rank greater than 2. [ABSTRACT FROM AUTHOR]

Gasull and Sabatini in [Ann. Mat. Pura Appl. 198 (2019), pp. 1985-2006] studied limit cycles of a Liénard system which has a fixed invariant curve, i.e., a Wilson polynomial Liénard system. The Liénard system can be changed into x = y − (x2 − 1)(x3 − bx), y = −x(1+y(x3 −bx)). For b ≤ 0.7 and b ≥ 0.76, limit cycles of the system are studied completely. But for 0.7 < b < 0 .76, the exact number of limit cycles is still unknown, and Gasull and Sabatini conjectured that the exact number of limit cycles is two (including multiplicities). In this paper, we give a positive answer to this conjecture and study all bifurcations of the system. Finally, we show the expanding of the moving limit cycle as b > 0 increases and give all global phase portraits on the Poincaré disk of the system completely. [ABSTRACT FROM AUTHOR]

Let Φ∞(d) denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree d. The sets Φ∞(d) are known for d ≤ 4. In this article we determine Φ∞(5) and Φ∞(6). [ABSTRACT FROM AUTHOR]

We introduce the class of everywhere like functions, which helps us to recover some known classes (such as that of everywhere surjective ones). We also study the algebraic genericity of this new class together with the class of fractal functions. [ABSTRACT FROM AUTHOR]

A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields. [ABSTRACT FROM AUTHOR]

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields. [ABSTRACT FROM AUTHOR]

In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface. In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations. [ABSTRACT FROM AUTHOR]

We prove a general form of Green's Formula and the Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane. We use VituSkin's localization of singularities method and a decomposition of a rectifiable curve in terms of a sequence of Jordan rectifiable sub-curves due to Carmona and Cufi. [ABSTRACT FROM AUTHOR]

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R3 and the hierarchy of positive, strongly quasi-positive, and quasipositive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions. [ABSTRACT FROM AUTHOR]

Hurwitz spaces are moduli spaces of curve covers. The isomorphism classes of covers of P¹ℂ with given ramification data are parameterized combinatorially by Nielsen tuples in the monodromy group G. The Artin braid group acts on Nielsen tuples, and the orbits of this action correspond to the connected components of the corresponding Hurwitz space. In this article we consider the case G = A5. We give a complete classification of the braid orbits for all ramification types, showing that the components are always distinguishable by the Fried-Serre lift invariant. [ABSTRACT FROM AUTHOR]

Hurwitz spaces are moduli spaces of curve covers. The isomorphism classes of covers of P¹ℂ with given ramification data are parameterized combinatorially by Nielsen tuples in the monodromy group G. The Artin braid group acts on Nielsen tuples, and the orbits of this action correspond to the connected components of the corresponding Hurwitz space. In this article we consider the case G = A5. We give a complete classification of the braid orbits for all ramification types, showing that the components are always distinguishable by the Fried-Serre lift invariant. [ABSTRACT FROM AUTHOR]

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curves of genus zero and one. Last, we give formulas for the graded Betti numbers for a class of curves of higher genus. [ABSTRACT FROM AUTHOR]

This note gives a proof that the A-polynomial of any nontrivial knot in S³ has nontrivial M-degree. [ABSTRACT FROM AUTHOR]

In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier codes since their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances. We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator. [ABSTRACT FROM AUTHOR]

Given a numerical semigroup H ⊆ (Z≥0, +), we consider the locus MHg,1 of smooth curves of genus g with a marked Weierstrass point of semigroup H. We show that for all semigroups H of genus g ≤ 6 the locus MHg,1 is irreducible and that for all but possibly two such semigroups it is stably rational. [ABSTRACT FROM AUTHOR]

The paper deals with amoebas of k-dimensional algebraic varieties in the complex algebraic torus of dimension n ≥ 2k. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of a k-dimensional algebraic variety in (C *)n, with n ≥ 2k, is finite. [ABSTRACT FROM AUTHOR]

In this paper we give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over Q with fixed modular parameterization, the Heegner points associated to (the maximal orders of) these quadratic imaginary fields are linearly independent. This extends results of Rosen and Silverman from the non-CM to the CM case. [ABSTRACT FROM AUTHOR]

We show that if C is a linearly normal smooth curve embedded by a line bundle of degree at least 2g + 3 + p, then the secant variety to the curve satisfies N3,p. [ABSTRACT FROM AUTHOR]

Kollár gave a series of examples of rational surfaces of Picard number 1 with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples has the same singularities as Kollár's examples. [ABSTRACT FROM AUTHOR]

J. Eells and L. Lemaire introduced k-harmonic maps, and Shaobo Wang showed the first variational formula. When k = 2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and we show the non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces and study k-harmonic curves in Euclidean spaces. Furthermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces. [ABSTRACT FROM AUTHOR]

Clunie and Hayman proved that if the spherical derivative ∥f'∥ of an entire function satisfies ∥f'∥(z) = O(∣z∣σ), then T(r, f) = O(rσ+1). We generalize this to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position. [ABSTRACT FROM AUTHOR]

Let C be a non-hyperelliptic curve of genus g. We recall some facts about curves endowed with a base-point-free g4¹. Then we prove that if the minimal degree of a surface containing the canonical model of C in Pkg-1 is g, then 7 ≤ g ≤ 12 and C carries exactly one g4¹. As a by-product, we deduce that if the canonical model of C in Pkg-1 is contained in a surface of degree at most g, then C is either trigonal or tetragonal or isomorphic to a plane sextic. [ABSTRACT FROM AUTHOR]

Max Noether's theorem asserts that if ω is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve, then the natural morphisms SymnH0(ω) → H0(ωn) are surjective for all n ⩾ 1. This is true for Gorenstein nonhyperelliptic curves as well. We prove that this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The results are independent and have different proofs. The first one is extrinsic, the second intrinsic. [ABSTRACT FROM AUTHOR]

For an odd integer N, we study the action of Atkin's U(2)-operator on the modular function x(t) associated to the Fermat curve: XN + YN = 1. The function x(t) is modular for the Fermat group φ(N), generically a noncongruence subgroup. If x(t) = q-1+Σ∞i=1 a(iN-1)qiN-1, we essentially prove that limn→0 a(n) = 0 in the 2-adic topology. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Yamauchi which states that the level $ N$ is $ \mathbb{Q}$ $ S^{new}(N,k)$ and level $ N.$ we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to $ d.$ [ABSTRACT FROM AUTHOR]

For $dgeq 2$, we construct a doubling measure $nu $ on $mathbb {R}^d$ and a rectifiable curve $Gamma $ such that $nu (Gamma )>0$. [ABSTRACT FROM AUTHOR]

By considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal. [ABSTRACT FROM AUTHOR]

We use the Aleksandrov reflection result of Chow and Gulliver to show that the emph {center of expansion} in expanding a given convex embedded closed curve $gamma _{0}subset mathbb {R}^{2}$ lies on a certain convex plane region interior to $gamma _{0}.$ [ABSTRACT FROM AUTHOR]

We define the notion of a {it positive path} of a homeomorphism of a topological space. It seems to be a natural object to understand Birkhoff's arguments in his proof of the celebrated Poincaré-Birkhoff theorem. We write the proof of this theorem, by using positive paths, and the proof of its generalization due to P. Carter. We will also explain the links with the free disk chains introduced in the subject by J. Franks. We will finish the paper by studying the local versions where the upper curve is not invariant and will explain why this curve or its image must be a graph to get such a generalization. [ABSTRACT FROM AUTHOR]

For primes $pequiv 1 pmod {12}$, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with $j$-invariant $frac {1728}{t}$ and values of a particular $_2F_1$-hypergeometric function over ${mathbb {F}_p}$. We also give a formula for traces of Hecke operators on spaces of cusp forms of weight $k$ and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's $tau $-function in terms of hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if $X$ is a quasi-complete intersection curve in $mathbb P^3$, then $I_X$ has a minimal free resolution $$ 0to oplus _{i=1}^{mu -3} S(d_{i+3}+c_1)to oplus _{i=1}^{2mu -4}S(-e_i)to oplus _{i=1}^mu S(-d_i)to I_Xto 0, $$ where $d_i,e_iin mathbb Z$ and $c_1=-d_1-d_2-d_3$. Therefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of $I_X$. Also we give a relation for the degrees of syzygy modules of $I_X$. Using this theorem, one can construct a smooth quasi-complete intersection curve $X$ such that the number of minimal generators of $I_X$ is $t$ for any given positive integer $tin mathbb Z^+$. [ABSTRACT FROM AUTHOR]

Given a finite sequence $U_{N}={u_{1},ldots ,u_{N}}$ of points contained in the $d$-dimensional unit torus, we consider the $L^{2}$ discrepancy between the integral of a given function and the Riemann sums with respect to translations of $U_{N}$. We show that with positive probability, the $L^{2}$ discrepancy of other sequences close to $U_{N}$ in a certain sense preserves the order of decay of the discrepancy of $U_{N}$. We also study the role of the regularity of the given function. [ABSTRACT FROM AUTHOR]

We obtain a Bernstein type result for entire two dimensional minimal graphs in $mathbb {R}^{4}$, which extends a previous result due to L. Ni. Moreover, we provide a characterization for complex analytic curves. [ABSTRACT FROM AUTHOR]

A covariant functor from the category of generic complex algebraic curves to a category of the $AF$-algebras is constructed. The construction is based on a representation of the Teichmüller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic $AF$-algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $mathbb {R}^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ mathbb {R}^n$ so that the canonical projection onto $ mathbb {T}^n $ is $ varepsilon $-dense. [ABSTRACT FROM AUTHOR]

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4$pi $. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than $4pi $ and which do not bound an orientable compact embedded minimal surface of genus greater than $g$, for any given $g$, is open in the $C^{2,alpha }$ topology. [ABSTRACT FROM AUTHOR]

We give a new proof, in scheme-theoretic language, of Tate's classical result on genus change of curves over imperfect fields in characteristic $p>0$. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus over the algebraic closure is divisible by $(p-1)/2$. [ABSTRACT FROM AUTHOR]

Let $C$ be a smooth projective algebraic curve of genus $q$ and $g$ an integer with $g\ge 4q+5$. For all integers $d\ge g-2q+1$ we prove the existence of a double covering $f:X\to C$ with $X$ a smooth curve of genus $g$ and the existence of a degree $d$ morphism $u:X\to \mathbb P^1$ that does not factor through $f$. By the Castelnuovo-Severi inequality, the result is sharp (except perhaps the bound $g\ge 4q+5$). [ABSTRACT FROM AUTHOR]

Assuming a subelliptic a-priori estimate we prove global analytic regularity for non-linear second order operators on a product of tori, using the method of majorant series. [ABSTRACT FROM AUTHOR]

This paper investigates the first part of Hilbert's 16th problem which asks about topology of the real projective algebraic curves. Using the Rokhlin-Viro-Fiedler method of complex orientation, we obtain new restrictions on the arrangements of ovals of projective algebraic curves of odd degree $d = 4k + 1$, $k \geq 2$, with nests of depth $k$. [ABSTRACT FROM AUTHOR]

Given $M(r,f)=\max_{|{z}|=r}\left( |{f(z)}|\right)$ , curves belonging to the set of points $\mathcal{M}=\left\{ z:|{f(z)}|=M(|{z}|,f)\right\}$ were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set $\mathcal{ M}$ could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions $f_1(z)$ and $f_2(z)$, if the maximum curve of $f_1(z)$ is the real axis, conditions are found so that the real axis is also a maximum curve for the product function $f_1(z)f_2(z)$ . By means of these results an entire function of infinite order is constructed for which the set $\mathcal{M}$ has an infinite number of isolated points. A polynomial is also constructed with an isolated point. [ABSTRACT FROM AUTHOR]

It is proved that the Menger universal curve $\mathcal{M}$ admits a continuous decomposition into pseudo-arcs with the quotient space homeomorphic to $\mathcal{M}$. [ABSTRACT FROM AUTHOR]

We study real algebraic morphisms from nonsingular real algebraic varieties $X$ with $\dim X \geq 1$ into nonsingular real algebraic curves $C$. We show, among other things, that the set of real algebraic morphisms from $X$ into $C$ is never dense in the space of all $\mathcal C^\infty$ maps from $X$ into $C$, unless $C$ is biregularly isomorphic to a Zariski open subset of the unit circle. [ABSTRACT FROM AUTHOR]

For all $g \ge 2$ there is a Riemann surface of genus $g$ whose automorphism group has order $8g+8$, establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large $g$ the Accola-Maclachlan surface was unique for $g= 0,1,2\mod 4$ and produced exactly one additional surface (the Kulkarni surface) for $g= 3\mod 4$. In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case. [ABSTRACT FROM AUTHOR]

In this paper, we give a bound for the number of rational points of a complete, geometrically irreducible, algebraic curve defined over a finite field. We compare it with other known bounds and discuss its sharpness. We also show that the asymptotic Drinfeld-Vladut bound can be generalized to the case of singular curves. [ABSTRACT FROM AUTHOR]

We give a simple proof of the following result of P. Carter: Given a twist homeomorphism of an annulus with at most one fixed point in the interior of the annulus, then there exists an essential simple closed curve inside this annulus meeting its image in at most the (possible) interior fixed point. [ABSTRACT FROM AUTHOR]