William N. Traves

ON THE KEYSPACE OF THE HILL CIPHER

In its most general form, the Hill ciphers keyspace consists of all matrices of a given dimension that are invertible over . Working from known results over finite fields, we assemble and prove a formula for the number of such matrices. We also compare this result with the total number of matrices and the number of involutory matrices for a given dimension and modulus, identifying the effects of change in dimension and modulus on the order of the keyspace.

Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture

Abstract.A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term sk of the independence sequence (s0,s1,…,sα) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent unimodality conjecture of Brown, Dilcher, and Nowakowski. We formulate the Roller-Coaster Conjecture to describe the possible linear orderings of terms of the independence sequence.

Differential operators on monomial rings

Rings of differential operators are notoriously difficult to compute. This paper computes the ring of differential operators on a Stanley-Reisner ring R. The D-module structure of R is determined. This yields a new proof that Nakai’s conjecture holds for Stanley-Reisner rings. An application to tight closure is described. @ 1999 Elsevier Science B.V. All rights reserved.

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi’s extension of Nakai’s conjecture to arbitrary characteristic. This gives a new proof of Nakai’s conjecture for curves and Stanley-Reisner rings.

Differential operators on orbifolds

An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarzs study of a map carrying invariant operators to operators on the orbifold and on an algorithm to compute rings of invariants using Grobner bases due to Derksen [Derksen, Harm, 1999. Computation of invariants for reductive groups. Adv. Math. 141 (2), 366-384]. It is also possible to avoid using Derksens algorithm, instead relying on the Reynolds operator and the Molien series.

Localization of the Hasse-Schmidt Algebra

The behaviour of the Hasse-Schmidt algebra of higher derivations under localization is stud- ied using Andr· e cohomology. Elementary techniques are used to describe the Hasse-Schmidt deriva- tions on certain monomial rings in the nonmodular case. The localization conjecture is then veried for all monomial rings.

From Pascal's Theorem to d-Constructible Curves

Abstract We prove a generalization of both Pascals Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve C of degree d, then the remaining k(k − d) points lie on a unique curve S of degree k − d. If S is a curve of degree k − d produced in this manner using a curve C of degree d, we say that S is d-constructible. For fixed degree d, we show that almost every curve of high degree is not d-constructible. In contrast, almost all curves of degree 3 or less are d-constructible. The proof of this last result uses the group structure on an elliptic curve and is inspired by a construction due to Möbius. The exposition is embellished with several exercises designed to amuse the reader.

Enumerative Algebraic Geometry of Conics

In this expository paper, we describe the solutions to several enumerative problems involving conies, including Steiners problem. The results and techniques presented here are not new; rather, we use these problems to introduce and demonstrate several of the key ideas and tools of algebraic geometry. The problems we discuss are the following: Given p points, / lines, and c conies in the plane, how many conies are there that contain the given points, are tangent to the given lines, and are tangent to the given conies? It is not even clear a priori that these questions are well-posed. The answers may depend on which points, lines, and conies we are given. Nineteenth and twentieth century geometers struggled to make sense of these questions, to show that with the proper interpretation they admit clean answers, and to put the subject of enumerative algebraic geometry on a firm mathematical foundation. Indeed, Hilbert made this endeavor the subject of his fifteenth challenge problem.

Invariant Theory and Differential Operators

Constructive invariant theory was a preoccupation of many nineteenth century mathematicians, but the topic fell out of fashion in the early twentieth century. In the latter twentieth century the topic enjoyed a resurgence, partly due to its connections with the construction of moduli spaces in algebraic geometry and partly due to the development of computational algorithms suitable for implementation in modern symbolic computation packages. In this survey paper we briefly discuss some of the history and applications of invariant theory and apply one particular algorithm that uses Gröbner bases to find invariants of linearly reductive algebraic groups acting on the Weyl algebra. After showing how we can present the ring of invariant differential operators in terms of generators and relations, we turn to the operators on the invariant ring itself. The theory is particularly nice for finite groups acting on polynomial rings, but we also compute an example involving an SL2C-action. In this example, we give a complete description of the generators and relations of D(G(2, 4)), the ring of differential

Quasi-Projective Varieties

We have developed the theory of affine and projective varieties separately. We now introduce the concept of a quasi-projective variety, a term that encompasses both cases. More than just a convenience, the notion of a quasi-projective variety will eventually allow us to think of an algebraic variety as an intrinsically defined geometric object, free from any particular embedding in affine or projective space.