Real Algebraic Curves Research Articles

Real symmetric matrices with partitioned eigenvalues

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions. For small matrices, we provide equations; for larger matrices, we explain how to use representation theory to find equations. We describe the ring of invariants under the action of the orthogonal group. For the subvariety of diagonal matrices, we give the degree.

Quadratic differentials of real algebraic curves

It was shown by J. C. Langer and D. A. Singer in their influential 2007 paper [5] that real algebraic curves can be interpreted as trajectories of meromorphic quadratic differentials defined on appropriate Riemann surfaces. The goal of this note is to suggest a more elementary approach to this problem that is based on the classical complex analysis. To demonstrate how our approach works, we apply it to the simplest non-trivial case of real algebraic curves; i.e. to conics.

Weak and semi normalization in real algebraic geometry

We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.

The Capacity of Quiver Representations and Brascamp–Lieb Constants

Abstract Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma $ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp–Lieb (BL) operator $T_{V,\sigma }$ associated to $(V,\sigma )$ and study its capacity, denoted by $\textbf{D}_Q(V, \sigma )$. When $Q$ is the $m$-subspace quiver, the capacity of quiver data is intimately related to the BL constants that occur in the $m$-multilinear BL inequality in analysis. We show that the positivity of $\textbf{D}_Q(V, \sigma )$ is equivalent to the $\sigma $-semi-stability of $V$. We also find a character formula for $\textbf{D}_Q(V, \sigma )$ whenever it is positive. Our main tool is a quiver version of a celebrated result of Kempf–Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties that carry information relevant to our main objects of study. It allows us to express the capacity of quiver data in terms of the character induced by $\sigma $ and sample points of the varieties involved. Furthermore, we use this character formula to prove a factorization of the capacity of quiver data. We also show that the existence of gaussian extremals for $(V, \sigma )$ is equivalent to $V$ being $\sigma $-polystable and that the uniqueness of gaussian extremals implies that $V$ is $\sigma $-stable. Finally, we explain how to find the gaussian extremals of a gaussian-extremisable datum $(V, \sigma )$ using the algebraic variety associated to $(V,\sigma )$.

An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in \({\mathbb {RP}}^2\) of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in \({\mathbb {CP}}^2\) of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.

V. A. Rokhlin and D. A. Gudkov Against the Background of Hilbert’s 16th Problem (According to Their Correspondence in 1971–1982)

The story of the friendship and collaboration between Vladimir Abramovich Rokhlin and the Nizhny Novgorod mathematician Dmitry Andreevich Gudkov during the last period of Rokhlin’s mathematical biography, when he worked in the topology of real algebraic varieties, in which he obtained remarkable results. The paper is based on the correspondence of 1971–1982 preserved in Gudkov’s archive containing 15 letters by V. A.Rokhlin and 8 letters by D. A. Gudkov.

HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

Planar polynomial Hamiltonian differential systems with global centers

In this paper, we give the necessary and sufficient 条件 for planar polynomial Hamiltonian systems to have a global center. This characterizes that the plane $\mathbb{R}^2$ is foliated by the level set of this polynomial Hamiltonian (that is a real algebraic curve), and each of the real algebraic curve is an oval (a compact closed branch). Furthermore, it is shown that for any natural number $n$ there exist planar polynomial Hamiltonian systems of degree $2n+1$ with a global center, which can exhibit one of the three types of centers: linear, nilpotent or degenerate, respectively, but planar polynomial Hamiltonian systems of degree $2n$ do not have global centers. In particular, the necessary and sufficient 条件 are givenfor planar cubic polynomial Hamiltonian systems to have a degenerate center.

Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Sigma of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of text {AdS}_4 space-time.

Real algebraic curves of bidegree (5,5) on the quadric ellipsoid

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our strategy is first to reduce to the construction of curves in the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, we combine different classical construction methods on toric surfaces, such as Dessin d'enfants and and Viro's patchworking method.

RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE

AbstractIn this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

On the Complexity of Computing the Topology of Real Algebraic Space Curves

This paper presents an algorithm to compute the topology of an algebraic space curve. This is a modified version of the previous algorithm. Furthermore, the authors also analyse the bit complexity of the algorithm, which is \(\widetilde{\cal O}\left( {{N^{20}}} \right)\), where N = max{d, τ}, d and τ are the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve. To our knowledge, this is the best bound among the existing work. It gains the existing results at least N2. Meanwhile, the paper contains some contents of the conference papers (CASC 2014 and SNC 2014).

Pull-back of singular Levi-flat hypersurfaces

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas -- Fernandez-Perez -- Mol.

On singularities of real algebraic sets and applications to kinematics

Abstract We address the question of identifying non-smooth points in V ℝ ( I ) {{\bf{V}}}_{{\mathbb{R}}}(I) the real part of an affine algebraic variety. Two simple algebraic criteria will be formulated and proven. As an application, we investigate the configuration spaces of the planar four-bar linkage and the delta robot and prove that all singularities are CS-singularities.

The lexicographic degree of the first two-bridge knots

We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.

The distance function from a real algebraic variety

For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpolyX,u(t2) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpolyX,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpolyX,u(t2)=EDpolyX∨,u(q(u)−t2) where X∨ is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X∪(X∨∩Q)∨.

Approximation by piecewise-regular maps

A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Rm. Let l be any nonnegative integer. We prove that every map of class Cl from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the Cl topology by piecewise-regular maps of class Ck, where k is an arbitrary integer satisfying k≥l. Next we derive consequences regarding algebraization of topological vector bundles.

Isotopic Meshing of a Real Algebraic Space Curve

This paper presents a new algorithm for computing the topology of an algebraic space curve. Based on an efficient weak generic position-checking method and a method for solving bivariate polynomial systems, the authors give a first deterministic and efficient algorithm to compute the topology of an algebraic space curve. Compared to extant methods, the new algorithm is efficient for two reasons. The bit size of the coefficients appearing in the sheared polynomials are greatly improved. The other is that one projection is enough for most general cases in the new algorithm. After the topology of an algebraic space curve is given, the authors also provide an isotopic-meshing (approximation) of the space curve. Moreover, an approximation of the algebraic space curve can be generated automatically if the approximations of two projected plane curves are first computed. This is also an advantage of our method. Many non-trivial experiments show the efficiency of the algorithm.

Integral closures in real algebraic geometry

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety: the regular functions and the continuous rational functions.