On amoebas of algebraic sets of higher codimension

The amoeba of a complex algebraic set is its image under the projection onto the real subspace in the logarithmic scale. We study the homological properties of the complements of amoebas for sets of codimension higher than 1. In particular, we refine A. Henriques’ result saying that the complement of the amoeba of a codimension k set is (k ? 1)-convex. We also describe the relationship between the critical points of the logarithmic projection and the logarithmic Gauss map of algebraic sets.     

The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.     

On the Rate of Quantum Ergodicity for Quantised Maps

We study the distribution of expectation values and transition amplitudes for quantised maps on the torus. If the classical map is ergodic then the variance of the distribution of expectation values will tend to zero in the semiclassical limit by the quantum ergodicity theorem. Similarly the variance of transition amplitude goes to zero if the map is weak mixing. In this paper we derive estimates on the rate by which these variances tend to zero. For a class of hyperbolic maps we derive a rate which is logarithmic in the semiclassical parameter, and then show that this bound is sharp for cat maps. For a parabolic map we get an algebraic rate which again is sharp. Submitted: May 31, 2008., Accepted: September 19, 2008.     

Data loci in algebraic optimization

We consider parametric optimization problems from an algebraic viewpoint. The idea is to find all of the critical points of an objective function thereby determining a global optimum. For generic parameters (data) in the objective function the number of critical points remains constant. This number is known as the algebraic degree of an optimization problem. In this article, we go further by considering the inverse problem of finding parameters of the objective function so it gives rise to critical points exhibiting a special structure. For example if the critical point is in the singular locus, has some symmetry, or satisfies some other algebraic property. Our main result is a theorem describing such parameters.     

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.     

Critical points of the integral map of the charged three-body problem

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.     

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

We construct a bijective continuous area preserving map from a class of elongated dipyramids to the sphere, together with its inverse. Then we investigate for which such solid polyhedrons the area preserving map can be used for constructing a bijective continuous volume preserving map to the 3D-ball. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids on the elongated dipyramids. In particular, we show that HEALPix grids can be obtained from these maps. We also study the optimality of the logarithmic energy of the configurations of points obtained from these grids.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

Isosingular loci of algebraic varieties

We define the notion of isosingular loci of algebraic varieties, following the analytic case first studied by Ephraim. These are subsets of points where the variety has a prescribed formal singularity type. We show that the isosingular loci of an algebraic variety are locally closed in the Zariski topology and the associated reduced subschemes are smooth. Moreover, assuming characteristic 0, we prove the existence of a decomposition of the formal neighborhoods at closed points into a product of the respective isosingular locus at that point and a smooth factor. One of the main obstructions in the positive characteristic case is the non-separability of the orbit map associated to the contact group, as first observed by Greuel and Pham for isolated singularities.     

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubi?s Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd₂(F[i])<∞, but not in general.     

The index of elliptic operators on manifolds with conical points

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the "eta" invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be nonzero. Moreover, we introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.     

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.     

Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.     

Geometric height inequalities and the Kodaira--Spencer map

We extend a result of Vojta on height inequalities for algebraic points oncurves over function fields to include the case of positivecharacteristic. The main tool used is the Kodaira--Spencer map anddestabilizing flags for vector bundles on curves.     

$\Omega$-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic. Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001     

Transcendency of local conjugacies in complex dynamics and transcendency of their values

Letp andq be polynomials of the same degree. A classical result of Böttcher says that there exists a functionf conformal in a neighborhood of infinity such thatf(p(z))=q(f(z)). We show thatf is transcendental and takes transcendental values at algebraic points unlessp andq are linearly conjugate to monomials or Chebychev polynomials. As an application, we show that the conformal map from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. A second application is the solution of a transcendency problem posed by Golomb.     

Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

Numerical Algorithms - In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on...     

A complex map with complex topology

We construct a simple example of a complex algebraic map, which (even locally) has an uncountable number of local topological types at points of the source space.