On arithmetical varieties of near-rings

This research was done during the second aurthor's stay in National Cheng-Kung University, Tainan. Support by the grant No. VRP92034 from the National Science Council of ROC is gratefully acknowledged.     

Congruence computations in principal arithmetical varieties

This paper is a continuation of the earlier paper by the same authors in which a primary result was that every arithmetical affine complete variety of finite type is a principal arithmetical variety with respect to an appropriately chosen Pixley term. The paper begins by presenting an extension of this result to all finitely generated congruences and, as an example, constructs a closed form solution formula for any finitely presented system of pairwise compatible congruences (the Chinese remainder theorem). It is also shown that in all such varieties the meet of principal congruences is also principal, and finally, if a minimal generating algebra of the variety is regular, it is shown that the variety is also regular and the join of principal congruences is again principal.     

Schubert polynomials and Bott-Samelson varieties

Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.?The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott-Samelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.?This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux. Received: November 27, 1997     

Schubert polynomials and classes of Hessenberg varieties

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.     

Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties

The aim of this paper is to prove that every finitely generated, arithmetical variety of finite type, in which every subdirectly irreducible algebra has linearly ordered congruences has a decidable first order theory of its finite members. The proof is based on a representation of finite algebras from such varieties by some quotients of special subdirect products in which sets of indices are partially ordered into dual trees. Then the result of M. O. Rabin about decidability of the monadic second order theory of two successors is applied.Presented by Stanley Burris.     

Mixed Hodge polynomials of character varieties

We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.     

Schubert polynomials and Arakelov theory of orthogonal flag varieties

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety ${\mathfrak X={\rm SO}_N/B}$ . We use these polynomials to describe the arithmetic Schubert calculus on ${\mathfrak X}$ . Moreover, we give a method to compute the natural arithmetic Chern numbers on ${\mathfrak X}$ , and show that they are all rational numbers.     

Kazhdan-Lusztig polynomials for certain varieties of incomplete flags

We give explicit formulas for the Kazhdan-Lusztig P- and R-polynomials for permutations coming from the variety F_1,n−1 of incomplete flags consisting of a line and a hyperplane.     

Positive polynomials on projective limits of real algebraic varieties

We reveal some important geometric aspects related to non-convex optimization of sparse polynomials. The main result, a Positivstellensatz on the fibre product of real algebraic affine varieties, is iterated to a comprehensive class of projective limits of such varieties. This framework includes as necessary ingredients recent works on the multivariate moment problem, disintegration and projective limits of probability measures and basic techniques of the theory of locally convex vector spaces. A variety of applications illustrate the versatility of this novel geometric approach to polynomial optimization.     

Quasi-smoothness and arithmetical surfaces

The paper of Villamayor (Math Z. 225 (1997) 317) introduced a notion of quasi-smoothness which makes it possible to associate a canonical tangent bundle to certain arithmetical schemes. Here, we study the behavior of this quasi-smoothness condition in the case of arithmetical regular surfaces.     

Prime-independent arithmetical functions

Summary This paper studies arithmetical functions that are prime-independent and multiplicative. Firstly, it establishes necessary and sufficient conditions for such functions to possess simple formulae relating them to the zeta function. Then it investigates asymptotic average-values and moments of such functions. The results apply to functions of ideals in algebraic numbers fields, or of isomorphism classes in certain categories, as well as to functions of positive integers. Entrata in Redazione il 10 ottobre 1972.     

The characteristic polynomials of abelian varieties of dimensions 3 over finite fields

We describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields.