二阶多项式系统固定奇点附近解的解析展开

研究二阶多项式系统所确定的通过固定奇点的解．利用牛顿图法研究了这种解在固定奇点处的幂级数展开式,给出有无穷多个解通过固定奇点的判据,并在只有有限个解通过固定奇点的情况下,给出了过固定奇点的解个数的上界．     

NURBS approximation of surface/surface intersection curves

We use a combination of both symbolic and numerical techniques to construct degree boundedC ^k -continuous, rational B-spline ε-approximations of real algebraic surface-surface intersection curves. The algebraic surfaces could be either in implicit or rational parametric form. At singular points, we use the classical Newton power series factorizations to determine the distinct branches of the space intersection curve. In addition to singular points, we obtain an adaptive selection of regular points about which the curve approximation yields a small number of curve segments yet achievesC ^k continuity between segments. Details of the implementation of these algorithms and approximation error bounds are also provided. Supported in part by NSF Grants CCR 92.22467, DMS 91-01424, AFOSR Grant F49620-10138 and NASA Grant NAG-1-1473. Supported in part by K.C. Wong Education Foundation, Hong Kong.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

Piecewise Rational Approximations of Real Algebraic Curves

1.IntroductionAnaJgebraicplanecurveCofdegreedinn2isimplicitlydefinedbyasinglepolynomialequationf(x,y)=Oofdegreedwithcoefficientsinn.Arationalalgebraiccurveofdegreedinn2canadditionaJlybedefinedbyrationalparametricequationswhicharegivenas(x=G1(u),y=G2(u)),whereG1andG2arerationalfunctionsinuofdegreed,i.e-,eachisaquotientofpolynomiaJ8inuofmtalmumdegreedwithcoefficientsinn.ffetionalcurvesaxeonlyasubsetofimplicitalgebraiccurvesofdegreed+1.Whi1eaJldegreetwocurves(conics)arerational,oIilyasubsetof…     

Blow-Up Estimates at Horizontal Points and Applications

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathéodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of C ^1,1 smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any C ^1,1 submanifold. Our technique also shows that C ² smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.     

Classification and higher order analysis of manipulator singularities

A purely algebraic approach to higher order analysis of (singular) configurations of rigid multibody systems with kinematic loops (CMS) is presented. Rigid body con.gurations are described by elements of the Lie group SE(3) and so the rigid body kinematics is determined by an analytical map f : V → SE(3), where V is the configuration space, an analytic variety. Around regular configurations V has manifold structure but this is lost in singular points. In such points the concept of a tangent vector space does not makes sense but the tangent space C_qV (a cone) to V can still be defined. This tangent cone can be determined algebraically using the special structure of the Lie algebra se (3), the generating algebra of the special Euclidean group SE (3), and the fact that the push forward map f_*, the tangential mapping C_qV → se (3), is given in terms of the mechanisms screw system. Moreover the differentials of f of arbitrary order can be expressed algebraically. The tangent space to the configuration space can be shown to be a hypersurface of maximum degree 4, a vector space for regular points. It is the structure of the tangent cone to V that gives the complete geometric picture of the configuration space around a (singular) point. Identification of the screw system and its matrix representation with the kinematic basic functions of the CMS allows an automatic algebraic analysis of mechanisms.     

The Space of Unordered Real Line Arrangements

The set of all unordered real line arrangements of given degree in the real projective plane is known to have a natural semialgebraic structure. The nonreduced arrangements are singular points of this structure. We show that the set of all unordered real line arrangements of given degree also has a natural structure of a smooth compact connected affine real algebraic variety. In fact, as such, it is isomorphic to a real projective space. As a consequence, we get a projectively linear structure on the set of all real line arrangements of given degree. We also show that the universal family of unordered real line arrangements of given degree is not algebraic.     

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.     

Global Structure of Planar Quadratic Semi-Quasi-Homogeneous Polynomial Systems

This paper study the planar quadratic semi-quasi-homogeneous polynomial systems(short for PQSQHPS). By using the nilpotent singular points theorem, blow-up technique, Poincaré index formula, and Poincaré compaction method, the global phase portraits of such systems in canonical forms are discussed. Furthermore, we show that all the global phase portraits of PQSQHPS can be-classed into six topological equivalence classes.     

A theory of cobordism for non-spherical links

We define an equivalence relation, called algebraic cobordism, on the set of bilinear forms over the integers. When , we prove that two 2n - 1 dimensional, simple fibered links are cobordant if and only if they have algebraically cobordant Seifert forms. As an algebraic link is a simple fibered link, our criterion for cobordism allows us to study isolated singularities of complex hypersurfaces up to cobordism. Received: August 24, 1995     

Matrix subspaces and determinantal hypersurfaces

Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.     

On singular formal deformations

We discuss the notion of singular formal deformation in algebraic setup. Such deformations show up in both finite and infinite dimensional structures. It turns out that there is a stronger version of singular deformation—called essentially singular—which arises from a singular curve in the base of the versal deformation.     

The Nother and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nother type theorems for Cμpiecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible Cμpiecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cμpiecewise algebraic curve is established.     

Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

The conformal geometry of regular hypersurfaces in the conformal space is studied.We classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to conformal equivalence.     

Projective Hypersurfaces with many Singularities of Prescribed Types

Patchworking of singular hypersurfaces is used to constructprojective hypersurfaces with prescribed singularities. Forall n 2, an asymptotically proper existence result is deducedfor hypersurfaces in Pⁿ with singularities of corank at most2 prescribed up to analytical or topological equivalence. Inthe case of T-smooth hypersurfaces with only simple singularities,the result is even asymptotically optimal, that is, the leadingcoefficient in the sufficient existence condition cannot beimproved, which is new even in the case of plane curves. Furthermore,an asymptotically proper existence result is proved for hypersurfacesin Pⁿ with quasihomogeneous singularities. The estimates substantiallyimprove all known (general) existence results for hypersurfaceswith these singularities.     

Projective invariants of CR-hypersurfaces

We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invariants for analytic hypersurfaces is given. The self-dual strongly ?-linearly convex hypersurfaces are characterized.     

Analytic continuation of resolvent kernels on noncompact symmetric spaces

For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.The author is a member of EDGE, Research Training Network, HPRN-CT-2000-00101, supported by the European Human Potential Programme.     

Actual and virtual dimension of codimension 2 general linear subspaces in $${\mathbb {P}}^n$$ P n

Geometriae Dedicata - In the paper we compute the virtual dimension (defined by the Hilbert polynomial) of a space of hypersurfaces of given degree containing s codimension 2 general linear...     

Degree matrices and divisibility of exponential sums over finite fields

By using the degree matrix, we provide an elementary and algorithmic approach to estimating the divisibility of exponential sums over prime fields, which improves the Adolphson–Sperber theorem obtained by using the Newton polyhedron. Our result also improves the Ax–Katz theorem on estimating the number of rational points on hypersurfaces over prime fields.