Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Lower Bounds on Representing Boolean Functions as Polynomials in $z_{m}$

Abstract: Define the $MOD_{m}$-degree of a boolean function $F$ to be the smallest degree of any polynomial $P$, over the ring of integers modulo $m$, such that for all 0-1 assignments $\vec{x}$, $F(\vec{x}) =0$ iff $P(\vec{x}) =0$. By exploring the periodic property of the binomial coefficients modulo $m$, two new lower bounds on the $MOD_{m}$-degree of the $MOD_{l}$ and $ eg MOD_{m}$ functions are proved, where $m$ is any composite integer and $l$ has a prime factor not dividing $m$. Both bounds improve from sublinear to $\Omega(n)$. With the periodic property, a simple proof of a lower bound on the $MOD_{m}$-degree with symmetric multilinear polynomial of the OR function is given. It is also proved that the majority function has a lower bound $n \over 2$ and the {MidBit} function has a lower bound $\sqrt{n}$.

Matrices of rational integers

Abstract: 1. Introduction. This subject is very vast and very old. It includes all of the arithmetic theory of quadratic forms, as well as many of other classical subjects, such as latin squares and matrices with elements + 1 or —1 which enter into Euler's, Sylvester's or Hadamard's famous conjectures. In recent years statistical research into block designs on one hand and research into finite projective geometries on the other hand have led to a large amount of progress in this area. Thirdly the possibility of help from high speed computers has raised new hopes and stimulated new research. Under our subject comes the whole of the theory of the unimodular group in n dimensions and that of the modular group with all its ramifications into number theory and function theory including complex multiplication. The study of space groups and parts of crystallography belongs to our subject also. Matrix theory is a natural part of algebra. However many difficult problems do not seem to yield easily to purely algebraic methods. I refer for instance to much modern research on eigen values. Either geometrical or analytical methods seem to be called to the rescue. On the other hand, much inspiration is obtained from the study of matrices with elements in a ring and not in a field. This sometimes brings out the finer nature of the theorems considered. So it seems that not only the methods of abstract algebra, but also those of analysis, geometry and number theory play an increasing role in matrix theory. This account is divided into several chapters. Each has its own bibliography which is not intended to be complete. In particular in the chapters concerned with classical material much of the older literature is completely ignored.

A Note on Height Pairings, Tamagawa Numbers, and the Birch and Swinnerton-Dyer Conjecture.

Abstract: Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak). Since different choices of lifting will change G(C,,) for only a finite number of v, G(Ak) is intrinsically defined independent of the choice of Cs-scheme structure. It may happen that G(k)cG(Ak) is discrete. This will be the case, for example, if G is affine. If so, we may try to compute the volume of G(Ak)/G(k ). Writing I F = residue field at v, q,,= ~IF,,, N,,= ,t~ GOF,,), the natural volume form gives Vol(G(C~))=N~q~_ ~ for all v6S. It can happen that [IN,,q~71 does not

Semistable models for modular curves of arbitrary level

Abstract: We produce an integral model for the modular curve $$X(Np^m)$$ over the ring of integers of a sufficiently ramified extension of $$\mathbf {Z}_p$$ whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of $$X(Np^m)$$ , which is a union of copies of a Lubin–Tate curve. In doing so we tie together non-abelian Lubin–Tate theory to the representation-theoretic point of view afforded by Bushnell–Kutzko types. For our analysis it was essential to work with the Lubin–Tate curve not at level $$p^m$$ but rather at infinite level. We show that the infinite-level Lubin–Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin–Tate spaces of finite level.

Diophantine Representation of Recursively Enumerable Predicates

Abstract: Publisher Summary This chapter presents a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients that devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. It is well-known that an algorithm for determining the solvability in integers would yield an algorithm for determining the solvability in positive integers and conversely. Hence, the solvability in positive integers is discussed. Lower-case Latin letters will always be variables whose range is the positive integers. Every recursively enumerable set of positive integers (e.g., the set of all prime numbers) coincides with the set of all positive values of some polynomial with integer coefficients.