On <Emphasis Type="Italic">n</Emphasis>-cubic Pyramid Algebras

In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n?1)-translation algebras. We also recover Iyama’s cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.     

Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

A resultant system as the set of coefficients of a single resultant

Explicit expressions for polynomials forming a homogeneous resultant system of a set of m+1 homogeneous polynomial equations in n+1<m+1 variables are given. These polynomials are obtained as coefficients of a homogeneous resultant for an appropriate system of n+1 equations in n+1 variables, which is explicitly constructed from the initial system. Similar results are obtained for mixed resultant systems of sets of n + 1 sections of line bundles on a projective variety of dimension n < m. As an application, an algorithm determining whether one of the orbits under an action of an affine irreducible algebraic group on a quasi-affine variety is contained in the closure of another orbit is described.     

Fast fully polynomial approximation schemes for minimizing completion time variance

We present fully polynomial approximation schemes (FPASs) for the problem of minimizing completion time variance (CTV) of a set of n jobs on a single machine. The fastest of these schemes runs in time O(n²/ε) and thus improves on all fully polynomial approximation schemes presented in the literature.     

On the L 2-extension of twisted holomorphic sections of singular hermitian line bundles

We prove a sharp Ohsawa–Takegoshi–Manivel type L ²-extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.     

A Tannakian Interpretation of the Elliptic Infinitesimal Braid Lie Algebras

Let n ≥?1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E ^# of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

The plancherel formula for the line bundles on SL(n + 1, ℝ)/S(GL(1,ℝ) × GL(n, ℝ))

In this paper we obtain the Plancherel formula for the spaces of L²-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, ?) and H = S(GL(1, ?) × GL(n 1, ?)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ_λ of the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

Gruppen secret sharing or how to share several secrets if you must?

Each member of an n-person team has a secret, say a password. The k out of n gruppen secret sharing requires that any group of k members should be able to recover the secrets of the other n ? k members, while any group of k ? 1 or less members should have no information on the secret of other team member even if other secrets leak out. We prove that when all secrets are chosen independently and have size s, then each team member must have a share of size at least (n ? k)s, and we present a scheme which achieves this bound when s is large enough. This result shows a significant saving over n independent applications of Shamir’s k out of n ? 1 threshold schemes which assigns shares of size (n ? 1)s to each team member independently of k. We also show how to set up such a scheme without any trusted dealer, and how the secrets can be recovered, possibly multiple times, without leaking information. We also discuss how our scheme fits to the much-investigated multiple secret sharing methods.     

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L _p -norm, 1≤p≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.     

Bundles, presemifields and nonlinear functions

Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF(2 ⁿ ), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p ⁿ , we derive bundles of functions over GF(p ⁿ ) of the form λ(x)*ρ(x), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p ² and give a representative of each. We compute all bundles of presemifields of orders p ⁿ ≤ 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived λ(x)*ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).     

Hasse–Witt invariants of symmetric complexes: an example from geometryInvariants de Hasse–Witt de complexes symétriques : un exemple géométrique

Jardine has defined Hasse–Witt invariants for symmetric bundles over schemes. This definition can be extended to symmetric complexes, that is symmetric objects in the derived category of bounded complexes of vector bundles over a scheme. In this Note we show how one can use these generalized invariants to give a neater proof of a comparison result on Hasse–Witt invariants of symmetric bundles attached to tame coverings of schemes. To cite this article: P. Cassou-Noguès et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 839–842.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Some moduli stacks of symplectic bundles on a curve are rational

Let C be a smooth projective curve of genus g?2 over a field k. Given a line bundle L on C, let Sympl2n,L be the moduli stack of vector bundles E of rank 2n on C endowed with a nowhere degenerate symplectic form up to scalars. We prove that this stack is birational to BGm×As for some s if deg(E)=n⋅deg(L) is odd and C admits a rational point P∈C(k) as well as a line bundle ξ of degree 0 with ξ⊗2?OC. It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

Existence of natural and projectively equivariant quantizations

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operators, and prove the existence of a quantization except when this parameter belongs to a discrete set of resonant values.