Affine Algebraic Curve Research Articles

Dembowski-Ostrom polynomials and Dickson polynomials

<p style='text-indent:20px;'>We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials.</p>

Characterization of Lipschitz normally embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is LNE in Cn if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in CPn with the property of its general affine traces being LNE in Cn. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

Approximation in the Mean on Rational Curves

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves.

Analytic and rational sections of relative semi-abelian varieties

The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme over an affine algebraic curve. These sections form a group; while the group of rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

An upper bound for the number of $S$-integral points on curves of genus zero

We consider affine plane algebraic curves of genus zero defined over number fields with three discrete valuations at infinity, and we determine an upper bound for the number of their $S$-integral points.

Dembowski–Ostrom polynomials and reversed Dickson polynomials

We discuss the problem of classifying Dembowski–Ostrom polynomials from the composition of reversed Dickson polynomials of arbitrary kind and monomials over finite fields of odd characteristic. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of all such Dembowski–Ostrom polynomials. Planar Dembowski–Ostrom polynomials have applications in many areas including cryptography and coding theory.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [12] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations.In this paper we show that, using the algebras F(E), one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order 5. Also, we prove a result announced in [13] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera type equations. Among the obtained algebras for equations considered in this paper and in [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus 1 and 0.

Carlitz–Wan conjecture for permutation polynomials and Weil bound for curves over finite fields

The Carlitz–Wan conjecture, which is now a theorem, asserts that for any positive integer n, there is a constant Cn such that if q is any prime power >Cn with GCD(n,q−1)>1, then there is no permutation polynomial of degree n over the finite field with q elements. From the work of von zur Gathen, it is known that one can take Cn=n4. On the other hand, a conjecture of Mullen, which asserts essentially that one can take Cn=n(n−2) has been shown to be false. In this paper, we use a precise version of Weil bound for the number of points of affine algebraic curves over finite fields to obtain a refinement of the result of von zur Gathen where n4 is replaced by a sharper bound. As a corollary, we show that Mullen's conjecture holds in the affirmative if n(n−2) is replaced by n2(n−2)2.

On residue maps for affine curves

We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.

A property of Hilbert curves of scrolls over surfaces

ABSTRACTLet (X,L) be a polarized manifold of dimension n. Its Hilbert curve is an affine algebraic plane curve of degree n encoding properties related to fibrations of X, defined by suitable adjoint linear systems to L. In particular, if (X,L) is a scroll over a smooth surface S, its Hilbert curve consists of n−2 parallel lines with a given slope and evenly spaced, plus a conic. Making its equation explicit, we show that this conic turns out to be itself the Hilbert curve of the ℚ-polarized surface , where ℰ is the rank-(n−1) vector bundle obtained by pushing down L via the scroll projection, if and only if ℰ is properly semistable in the sense of Bogomolov.

HC-equivalence vs numerical equivalence for ample line bundles on surfaces

Given a smooth complex projective variety X and an ample line bundle \(\mathcal{L}\) on it, one can associate the Hilbert curve with \((X,\mathcal{L})\). This is a plane affine algebraic curve, whose equation depends only on the numerical characters of X and \(\mathcal{L}\). In particular, two numerically equivalent ample line bundles on X lead to the same Hilbert curve. Focusing on the case of surfaces, in this paper we investigate how far the converse is from being true.

Canonical Projective Embeddings of the Deligne–Lusztig Curves Associated to2A2,2B2, and2G2

Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to 2 A 2 , 2 B 2 and 2 G 2 Daniel M. Kane May 15, 2015 Abstract The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2 A 2 , 2 B 2 and 2 G 2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for d the Coxeter number of these groups, we find polynomials for each of these models that cut out the F q -points, the F q d -points and the F q d+1 - points, and demonstrate a relation satisfied by these polynomials. Introduction There are four kinds of finite groups of Lie type of rank 1. The associated Deligne-Lusztig varieties for the Coxeter classes of these groups all give affine algebraic curves. The completions of these curves have several applications in- cluding the representation theory of the associated group ([1, 5]), coding theory ([4]) and the construction of potentially interesting covers of P 1 ([3]). In this paper, we consider these curves associated to the groups G = 2 A 2 , 2 B 2 and 2 G 2 . The remaining curve is associated to G = A 1 and is P 1 , but we do not cover this case as it is easy and doesn’t follow many of the patterns found in the analysis of the other three cases. For each of these curves, we explicitly construct an embedding C ,→ P(W ) where W is a representation of G of dimension 3,5, or 14 respectively, and provide an explicit system of equations cutting out C. The curve associated to 2 A 2 is the Fermat curve. The curve associated to B 2 is also well-known though not immediately isomorphic to our embedding. Embeddings of the curve associated to 2 G 2 were not known until recently. In [6], they constructed an explicit curve with the correct genus, symmetry group and number of points. Later, in [2], Eid and Duursma use this description to independently arrive at the embedding we produce in this paper. ∗ University of California, San Diego, Department of Mathematics / Department of Computer Science and Engineering, 9500 Gilman Drive #0404, La Jolla, CA 92093 dakane@ucsd.edu

Zero sets of bivariate Hermite polynomials

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m consists of exactly n+m disjoint branches and possesses n+m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R2, are completely different for the three families analyzed.

Congruence kernels around affine curves

Abstract Let S be a smooth affine algebraic curve, and let S ˚ ${\text{{$\mathring{S}$}}}$ be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of mapping class groups by showing that the congruence kernel ker Mod ( S ˚ ) ^ → Out ( π 1 ( S ˚ ) ^ ) $ \ker \Bigl (\widehat{\mathrm {Mod}(\mathring{S})} \rightarrow \mathrm {Out}(\widehat{\pi _1(\mathring{S})})\Bigr ) $ lies in the centralizer of every braid in Mod ( S ˚ ) ${{\mathrm {Mod}(\text{{$\mathring{S}$}})}}$ . As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex 𝒞 ˇ ( S ˚ ) ${\check{\mathcal {C}}(\text{{$\mathring{S}$}})}$ when S is affine, and a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.

An Explanation to “Rolling Simplexes and Their Commensurability” (Field Equations in Accordance with Tycho Brahe)

Various Cartesian models of central power fields with quadratic dynamics are studied. These examples lead the reader to comprehension of basic aspects of the differential algebraic-geometrical Brahe–Descartes–Wotton theory, which embraces central power fields whose dynamics is composed of flat affine algebraic curves of degree at most N (N = 1, 2, 3, . . .). When N = 2, a quadratic rolling simplex law is proved by purely algebraic means.

Laws of rolling simplexes (feild equations according to Tycho Brahe)

Models of central power fields with quadratic dynamics suggested by Rene Descartes are studied. These examples gradually lead the reader to the understanding of base notions of differen-tial algebraic-geometric theory of Brahe-Descartes-Wotton including central fields whose dynamics is formed by planar affine algebraic curves of degree not exceeding N (N = 1, 2, 3,...).

Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

The parameter space S p \mathcal {S}_p for monic centered cubic polynomial maps with a marked critical point of period p p is a smooth affine algebraic curve whose genus increases rapidly with p p . Each S p \mathcal {S}_p consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of S p \mathcal {S}_p , and of its smooth compactification, in terms of these escape regions. In particular, it computes the Euler characteristic. It concludes with a discussion of the real sub-locus of S p \mathcal {S}_p .

Semi-group Conditions for Affine Algebraic Plane Curves with More than One Place at Infinity

An interesting and open question is the classification of affine algebraic plane curves. Abhyankar and Moh [1] completely described the possible links at infinity for those curves where the link has just one component, a knot. Such curves are said to have one place at infinity. The Abhyankar-Moh result has been of great assistance in classifying those polynomials which define a connected curve with one place at infinity. This paper provides a new proof of the Abhyankar-Moh result which is then used to find a description for the case where the polynomial defines a curve with one point at infinity.

ON THE ISOMORPHISM PROBLEM FOR RINGS OF DIFFERENTIAL OPERATORS OVER AFFINE CURVES#

ABSTRACT In the group Aut k (A 1(k)) of k-automorphisms of A 1: = A 1(k) (the first Weyl algebra on a field k of any characteristic p ≥ 0), we solve the following problem: Find σ ∈ Aut k (A 1), such that where 0 ≠ b ∈ k[t] and n = deg(b). This problem is a particular case of the general problem of Stafford (1987) on isomorphisms between two k-algebras ±b𝒟 and ±b𝒟′ both Morita equivalent to A 1. In this paper, we study affine algebraic curves X(b) introduced by Letzter (1992) and Perkins (1991) and their algebra of differential operators ±b𝒟(X(b)). Due to the resolution of the problem above, we find the condition to have an isomorphism between two such algebras of differential operators. In the case of isomorphism, we define an explicit isomorphism. In particular, we make explicit isomorphisms announced in Letzter (1992) and Perkins (1991). Notice that in case k is an algebraically closed field of characteristic zero, the class of algebras of differential operators ±b𝒟(X(b)) is a very important one, since any k-algebra ±bℬ Morita equivalent to A 1 is isomorphic to some ±b𝒟(X(b)) in Kouakou (2003). Dans le groupe Aut k (A 1(k)) des k-automorphismes de A 1 = A 1(k) (la première algèbre de Weyl sur k, un corps commutatif de caractéristique quelconque bip) nous résolvons le problème: Trouver σ ∈ Aut k A 1 tel que σ (A 1 + b) = A 1 + t n où 0 ≠ b ∈ k[t] et n = deg(b). Ce problème est un cas particulier du problème général de Stafford (1987) sur l’existence d’isomorphisme entre deux k-algèbres ±b𝒟 et ±b𝒟′ toutes les deux Morita équivalentes à A 1. Dans ce papier, nous étudions les courbes affines X(b) introduites par Letzter (1992) et Perkins (1991) et leurs algèbres d’opérateurs diffrentiels associés ±b𝒟(X(b)). Graˆce á la résolution du problème ci-dessus, nous trouvons la condition pour que deux quelconques de ces algèbres soient isomorphes. Dans le cas d’isomorphisme, nous donnons explicitement un isomorphisme. En particulier, nous explicitons les isomorphismes annoncés dans Letzter (1992) et Perkins (1991). Notons que dans le cas où k est un corps algébriquement clos de caractéristique nulle, la classe des algèbres d’opérateurs differentiels ±b𝒟(X(b)) est très importante puisque toute k-algèbre ±bℬ Morita équivalente à A 1 est isomorphe à un ±b𝒟(X(b)); see Kouakou (2003).

ON THE CURVATURE OF THE REAL MILNOR FIBER

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$ . The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$ : $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$ .