Howe Pairs in the Theory of Vertex Algebras

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.Comment: A few typos corrected, final versio

Note on Integer Factoring Methods IV

This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.Comment: 20 Pages, New Versio

Using character varieties: Presentations, invariants, divisibility and determinants

If G is a finitely generated group, then the set of all characters from G into a linear algebraic group is a useful (but not complete) invariant of G . In this thesis, we present some new methods for computing with the variety of SL2C -characters of a finitely presented group. We review the theory of Fricke characters, and introduce a notion of presentation simplicity which uses these results. With this definition, we give a set of GAP routines which facilitate the simplification of group presentations. We provide an explicit canonical basis for an invariant ring associated with a symmetrically presented group\u27s character variety. Then, turning to the divisibility properties of trace polynomials, we examine a sequence of polynomials rn(a) governing the weak divisibility of a family of shifted linear recurrence sequences. We prove a discriminant/determinant identity about certain factors of rn( a) in an intriguing manner. Finally, we indicate how ordinary generating functions may be used to discover linear factors of sequences of discriminants. Other novelties include an unusual binomial identity, which we use to prove a well-known formula for traces; the use of a generating function to find the inverse of a map xn ∣→ fn(x); and a brief exploration of the relationship between finding the determinants of a parametrized family of matrices and the Smith Normal Forms of the sequence

Solving Commutative Relaxations of Word Problems

We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the commutative word problem, referred to as the “Zero-to-All reachability” problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples

Gr\"obner-Shirshov bases for categories

In this paper we establish Composition-Diamond lemma for small categories. We give Gr\"obner-Shirshov bases for simplicial category and cyclic category.Comment: 20 page

Doctor of Philosophy

dissertationAbstraction plays an important role in digital design, analysis, and verification, as it allows for the refinement of functions through different levels of conceptualization. This dissertation introduces a new method to compute a symbolic, canonical, word-level abstraction of the function implemented by a combinational logic circuit. This abstraction provides a representation of the function as a polynomial Z = F(A) over the Galois field F2k , expressed over the k-bit input to the circuit, A. This representation is easily utilized for formal verification (equivalence checking) of combinational circuits. The approach to abstraction is based upon concepts from commutative algebra and algebraic geometry, notably the Grobner basis theory. It is shown that the polynomial F(A) can be derived by computing a Grobner basis of the polynomials corresponding to the circuit, using a specific elimination term order based on the circuits topology. However, computing Grobner bases using elimination term orders is infeasible for large circuits. To overcome these limitations, this work introduces an efficient symbolic computation to derive the word-level polynomial. The presented algorithms exploit i) the structure of the circuit, ii) the properties of Grobner bases, iii) characteristics of Galois fields F2k , and iv) modern algorithms from symbolic computation. A custom abstraction tool is designed to efficiently implement the abstraction procedure. While the concept is applicable to any arbitrary combinational logic circuit, it is particularly powerful in verification and equivalence checking of hierarchical, custom designed and structurally dissimilar Galois field arithmetic circuits. In most applications, the field size and the datapath size k in the circuits is very large, up to 1024 bits. The proposed abstraction procedure can exploit the hierarchy of the given Galois field arithmetic circuits. Our experiments show that, using this approach, our tool can abstract and verify Galois field arithmetic circuits up to 1024 bits in size. Contemporary techniques fail to verify these types of circuits beyond 163 bits and cannot abstract a canonical representation beyond 32 bits

Finite polynomial maps and G-variant map germs.

The first half of this thesis is devoted to the study of finite polynomial maps en --4 en and the use of Grobner bases to determine if a given map is finite. We begin by examining those maps which have quasihomogeneous components, and give a simple condition for such maps to be finite. This condition is extended to those maps which are quasihomogeneous as above, but with extra lower order terms. Next, we give a general criterion for testing the finiteness of a given polynomial map and an implementation in the Maple computer algebra system. Our next step is to generalize our results to regular maps between affine varieties. Again, a finiteness criterion is given, plus its implementation in Maple. Lastly in this half, we consider the trace bilinear form associated with a finite map and show how it may be used to find real roots of a polynomial system. The second half of the thesis is concerned with the study of G-variant map germs, which commute with the action of a finite group G on the source and target spaces. We give a relation between the G-variant degree associated with a map germ, bilinear forms on the local algebra and preimages of zero under a perturbation of the original map. We look at both the complex and real affine space situation. We then give the equivalent results when we do not have a 'good' deformation of the map, when we have two groups acting and when we use modular representations. Next, we give an invariant of G-variant maps which is stronger than G-degree, based upon a lattice of vector subspaces. Finally, we examine the structure of the class of G-variant maps and consider criteria for maps to have 'good' deformations and to be finite. We then give ways of determining generators for the class of maps by generalizing theorems of Noether and Molien

Algebraic and Geometric Properties of Hierarchical Models

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models