Algebraic density property of homogeneous spaces

Let X be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that X is equipped with several fixed point free nondegenerate SL₂-actions satisfying some mild additional assumption. Then we prove that the Lie algebra generated by completely integrable algebraic vector fields on X coincides with the space of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form G/R where G is a linear algebraic group and R is a closed proper reductive subgroup of G.     

Algebraic Shifting of Finite Graphs

In the present article, for bipartite graphs and chordal graphs, their exterior algebraic shifted graph and their symmetric algebraic shifted graph are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that for a ≥ 3 and b ≥ 3, the exterior algebraic shifted graph of the complete bipartite graph K _a,b of size a, b is different from the symmetric algebraic shifted graph of K _a,b . Second, we will show that the exterior algebraic shifted graph of any chordal graph G coincides with the symmetric algebraic shifted graph of G. In addition, it will be shown that the exterior algebraic shifted graph of any chordal graph G is equal to some combinatorial shifted graph of G.     

A construction of wavelets

In this note, given a multiresolution analysis, we construct a class of four-coefficient scaling filters using only elementary algebraic operations. The method of construction also reveals that the sum conditions ∑a _even=∑a _odd=1 can also be verified without referring to the vanishing moment property.     

Algebraic Independence by Approximation Method (II)

Let f (x) be a continued fraction with elements a _n x, where coefficients a _n are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α₁,...,α_s with distinct absolute values the algebraic independence of the values f(α₁), ..., f(α_s) is proved under certain assumption concerning only with a _n . For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈ℤ) is also established. Received March 27, 1998, Accepted September 28, 1998     

Isotropic Trialitarian Algebraic Groups

By modifying a construction from Knuset al., we construct all isotropic algebraic groups of type³D₄and⁶D₄over an arbitrary field of characteristic ≠ 2. We also provide a nice isomorphism criterion for such groups. The results of this paper extend the main results of Allison (using entirely different methods) to fields of nonzero characteristic and algebraic groups.     

Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions =[a ₀;a ₁,a ₂,...] is proved under growth conditions involving the denominatorsq _n of the convergents and shifted partial quotientsa _n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.     

On Dynamic Algorithms for Algebraic Problems

In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, iff(x₁,x₂,…,x_n) = (y₁, y₂,…,y_m) is an algebraic problem, we consider answering on-line requests of the form “change inputx_ito valuev” or “what is the value of outputy_j?” We first present lower bounds for some simply stated algebraic problems such as multipoint polynomial evaluation, polynomial reciprocal, and extended polynomial GCD, proving an Ω(n) lower bound for the incremental evaluation of these functions. In addition, we prove two time-space trade-off theorems that apply to incremental algorithms for almost all algebraic functions. We then derive several general-purpose algorithm design techniques and apply them to several fundamental algebraic problems. For example, we give antime per request algorithm for incremental DFT. We also present a design technique for serving incremental requests using a parallel machine, giving a choice of either optimal work with respect to the sequential incremental algorithm or superfast algorithms withO(log log n) time per request with a sublinear number of processors.     

Generalized Poincaré Classes and Cubic Equivalences

Let Y be a smooth projective algebraic surface over ?, and T(Y) the kernel of the Albanese map CH₀(Y)_deg0 → Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E₁ × E₂, such as an abelian or a Kummer surface, then one can easily construct an abelian variety B and a surjective 'regular' homomorphism B^?z2 → T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P⁴. This implies a sort of boundedness result for T(Y). It is natural to ask if this is the case for any smooth projective algebraic surface Y ? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 (2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k—cycles on any projective algebraic manifold X, and arrive at, from a conjectural standpoint, necessary and sufficient cohomological conditions on X for which the question can be answered affirmatively.     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

Coxeter–Petrie Complexes of Regular Maps

Coxeter–Petrie complexes naturally arise as thin diagram geometries whose rank 3 residues contain all of the dual forms of a regular algebraic map M. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric groupS₃ , giving in all six dual forms, and whose source is the outer automorphism group of the infinite triangle group generated by involutions s₁, s₂, s₃, subject to the additional relation s₁s₃ =  s₃s₁. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s₁,s₃ , s₁s₃. These involutions together with the involutions₂ can be taken to define the nodes of a Coxeter diagram of shape D₄(with the involution s₂at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank 3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map.     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

Dimension Theory and Nonstable K 1

This article provides an introduction to A. Bak's theory of group-valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K ₁ and K ₂.     

Algebraic K-Theory and the Conjectural Leibniz K-Theory

The analogy between algebraic K-theory and cyclic homology is used to build a program aiming at understanding the algebraic K-theory of fields and the periodicity phenomena in algebraic K-theory. In particular, we conjecture the existence of a Leibniz K-theory which would play the role of Hochschild homology. We propose a motivated presentation for the Leibniz K ₂-group ofa field.     

Algebraic geometry over the residue field of the infinite place

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 𝔽_∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.     

Standard P3-bundles of exponent two on algebraic surfaces

Let V_K be a twisted form of P³ over the function field of an algebraic surface with Pic V_K generated by the half of the canonical line bundle. We construct an algebraic fibre space V → X projective flat over a smooth projective surface X with the generic fibre V _K → Spac K satisfying some properties.     

The Volodin model for hermitian algebraic K-theory

We define the Volodin hermitian algebraic K-theory for a (discrete) ring with an involution and show that it is isomorphic to Karoubi's hermitian algebraic K-theory. We also construct the Volodin model X(R _*) of hermitian algebraic K-theory for a simplicial ring R _* and show that it is a homotopy fiber of the map B Ô(R _*)B Ô(R _*)⁺. We also prove the general linear version of this result, which has been claimed in the existing literature, but whose proof was overlooked.     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.