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Showing 1 to 16 of 16 for “"Polynomial rings"”.
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Covering Systems of Polynomial Rings Over Finite Fields
In 1950 Paul Erdos observed that every integer belonged to a certain system of congruences with distinct moduli. He called such systems of congruences covering systems. Utilizing his covering system, he disproved a conjecture of de Polignac asking, “for every odd k, is there a prime of the form 2n …
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Ideal Membership in Polynomial Rings Over the Integers
The approach to the ideal membership problem for Z [X] followed here is based on some properties (such as Weierstrass Division) of the ring Z p⟨X⟩ of restricted power series with coefficients in the ring Z p of p-adic integers. We also consider the ideal membership problem for ideals of …
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Differential Polynomial Rings: Order Properties and Morita Equivalence
… where the latter is the ring of differential polynomials. Three properties are considered.
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Valued Difference Fields as Modules Over Twisted Polynomial Rings
More generally, the thesis considers arbitrary valued fields instead of (k((t)), v), with the Frobenius map replaced by an arbitrary self-embedding of the valued field. Many results go through in this more general setting with some additional assumptions.
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Powers of Monomial Ideals
We study monomial ideals in polynomial rings in two variables x, y over a field K. We determine various monomial ideals I such that Ik = (Mn)k where M is the maximal ideal generated by x, y and k is the least such integer. This is related to the well-known notion of Ratliff-Rush closures of an …
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Gröbner basis techniques for certain problems in coding and systems theory
… development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as …
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Elliptic Curves Over Finite Fields
… In Chapter 3 we define derivations on arbitrary polynomial rings, and prove the group law for elliptic curves. Chapter 4 discusses elliptic curves over finite fields and proves some results on counting points.
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Álgebra de corpos finitos aplicada à teoria da codificação: estudo do codificador BCH.
… fields and are characterized by operations on polynomials. In order to understand the operation of the BCH codes, we will make a brief introduction to the study of rings, fields, congruence relations on the integers, polynomial rings, Galois fields, and vector spaces. Next, we’ll be introduced …
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Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras
… second is by certain classes on bimodules over polynomial rings, called Soergel bimodules, and the third is by certain categories of constructible sheaves on the affine flag manifold (for the Langlands dual group). We prove results relating all three of these categorifications, and use them to …
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On Groebner Bases of (Non)commutative Free Algebras
The set of all polynomials in a collection of variables with coefficients in a given field is an important mathematical object, called the polynomial ring. The primary objects of study in this theory are sets of polynomials that contain additional mathematical structure called ideals. The theory of …
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Unique Signed Minimal Wiring Diagrams and the Stanley-Reisner Correspondence
… rely on the primary decomposition of ideals in polynomial rings.</p> <p>Stanley-Reisner theory provides a one-to-one correspondence between squarefree monomial ideals and abstract simplicial complexes. In this work, we use this correspondence to determine conditions under which a given set of …
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Verificación formal en ACL2 del algoritmo de Buchberger
… computation in ACL2 in which: (1) Multivariate polynomial rings are formalized. This formalization is abstract: it encapsulates a coefficient ring which is used for the construction of polynomials and the verification of their properties. (2) These rings are equipped with an ordering relation …
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A Computational Approach to the Quillen-Suslin Theorem, Buchsbaum-Eisenbud Matrices, and Generic Hilbert-Burch Matrices
… of modules of low projective dimension over polynomial rings whose free resolutions have known special structure. We begin with projective modules and investigate a computational approach to a famous theorem of Quillen-Suslin, which states that every finitely generated projective module over …
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An Introduction to Hilbert’s Nullstellensatz: an Insight for the Algebraically Minded
… groups and homomorphisms and then continues onto rings, ideals, radicals and quotient rings and relations between them. Some additional theorems are marked with a star to denote that the statement is not necessary for the algebraic proof of Hilbert's Nullstellensatz, but it might be necessary for …
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Polynomial Models for Systems Biology: Data Discretization and Term Order Effect on Dynamics
… we consider the modeling framework of polynomial dynamical systems over finite fields constructed from experimental data. We present and propose solutions to two problems inherent in this modeling method: the necessity of appropriate discretization of the data and the selection of a …