The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.     

Algebra-geometry of piecewise algebraic varieties

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.     

Recent researches on multivariate spline and piecewise algebraic variety

The multivariate splines as piecewise polynomials have become useful tools for dealing with Computational Geometry, Computer Graphics, Computer Aided Geometrical Design and Image Processing. It is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials. Recently the surfaces are mainly represented by multivariate piecewise polynomials (i.e. multivariate splines), so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry. Moreover, the piecewise algebraic variety will be also important in computational geometry, computer graphics, computer aided geometrical design and image processing. The purpose of this paper is to introduce some recent researches on multivariate spline, piecewise algebraic variety (curve), and their applications.     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

Pure homology of algebraic varieties

We show that for a complete complex algebraic variety the pure term of the weight filtration in homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce image homology for noncomplete varieties.     

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.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

THE DIMENSION OF A CLASS OF BIVARIATE SPLINE SPACES

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N(o)ther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

PIECEWISE SEMIALGEBRAIC SETS

Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of Rn satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of C^μ piecewise semialgebraic sets are also discussed.     

The Bezout number of piecewise algebraic curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.     

Strictly pseudoconvex domains and algebraic varieties

In the paper, we consider applications of strictly pseudoconvex domains to the problems of algebraicity and rationality. We give a new proof of the Kodaira theorem on the algebraicity of a surface and we also prove a multidimensional version of this theorem. Theorems analogous to the Hodge index theorem and the Lefschetz theorem about (1, 1)-classes are obtained for strictly pseudoconvex domains. Conjectures on the geometry of strictly pseudoconvex domains on algebraic surfaces are formulated. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 414–422, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the International Science Foundation under grant No. 508.     

零维代数簇的分解及其应用

本文利用Groebner基，给出了一种分解零维代数簇的方法，并且讨论了这种方法在理想的准素分解以及几何定理机器证明中的应用．     

Projective push-forwards in the Witt theory of algebraic varieties

We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas.     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

Fuzzy sets and type 2 under algebraic product and algebraic sum

The concept of fuzzy sets of type 2 has been proposed by L.A. Zadeh as an extension of ordinary fuzzy sets. A fuzzy set of type 2 can be defined by a fuzzy membership function, the grade (or fuzzy grade) of which is taken to be a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1].This paper investigates the algebraic properties of fuzzy grades (that is, fuzzy sets of type 2) under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring. Moreover, the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well-known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join, meet and algebraic product.     

The algebraic dimension of compact complex threefolds with vanishing second Betti number

This note investigates compact complex manifolds X of dimension 3 with second Betti number b₂(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b₁(X) = 1 and b₃(X) = 0 or that b₁(X) = 0 and b₃(X) = 2. The main idea is to show that c₃(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S⁶ cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b₂(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X P₁ is holomorphic.     

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

The group of R-equivalence classes for all adjoint semisimple classical algebraic groups is computed. Examples of stably non-rational adjoint simple groups of type D _n ,n≥3, are presented. The complete stable birational classification of adjoint simple groups of rank 3 is given. This paper was written while the author was on sabbatical leave at the Université catholique de Louvain. Typeset byA _M S-T_EX