On different notions of tameness in arithmetic geometry

The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.     

A non-canonical approach to arithmetic spin geometry and physical applications

A basic requirement of adelic physics is the principle of invariance of the fundamental physical laws under a change of the underlying number field proposed by I.V. Volovich (cf. [20]). In this paper, we develop a manifestly number field invariant approach to Yang-Mills theory, which is formulated within the framework of arithmetic geometry. As well source fields as the Higgs mechanism are incorporated. For this purpose a non-canonical approach to arithmetic spin geometry is proposed, and its physical applications are analyzed. The associated bundle construction is performed in the setting of arithmetic geometry. Furthermore the arithmetic analogue of the following well-known differential geometric fact is proven: Every covariant derivation on a torsor induces a canonical covariant derivation on the associated object.     

The arithmetic genus and extrinsic geometry of smooth projective variety

Some theorems concerning the projective normality and extrinsic geometry on the smooth variety are proved under some supposed conditions related to the arithmetic genus of the smooth variety.     

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.     

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.     

Implementing Taylor models arithmetic with floating-point arithmetic

The implementation of Taylor models arithmetic may use floating-point arithmetic to benefit from the speed of the floatingpoint implementation. The issue is then to take into account the roundoff errors. Here, we assume that the floating-point arithmetic is compliant with the IEEE-754 standard. We show how to get tight bounds of the roundoff errors, and more generally how to get high accuracy for the coefficients as well as for the bounds on the roundoff errors     

Primes in arithmetic progression

A method for a computer search of primes in arithmetic progression is described. Six sequences of length 16 and 21 sequences of length 15 were found as well as numerous sequences of lengths 13 and 14.     

Algorithms in unnormalized arithmetic

Calculation in unnormalized arithmetic provides, as has been described elsewhere, an estimate of the error of each quantity, initial, intermediate, and final, in a calculation. In an acceptable computational algorithm, this error estimate should be close to the true probable error. (The probable error is defined as follows: each computed quantity is defined, in the statement of the problem, explicitly or implicitly as some function of the input data, which are however known only up to errors having some distribution. As the input data vary in their joint distribution, the value of this function varies in a distribution whose variance determines the probable error of the result.) Many computational algorithms fail to achieve this goal, because whenever two numbers are combined arithmetically (+, -, ×, or ÷) in a computer, anycorrelation of the errors of the two numbers is perforce ignored; in consequence, the error estimate of a computed result may be greater than, comparable with, or smaller than the probable error, in which case the algorithm is called conservative, faithful, or liberal, respectively.In this article, a matrix-inversion algorithm based on an assignment strategy is described, which is very close to faithful, as evidenced by extensive numerical tests, which are also described. A test matrix has elements with varying numbers of leading zeros, the remaining bits in a 43-bit pattern are regarded as meaningful; its inverse is computed. A series of perturbed matrices are formed by adding or subtracting a one-bit in the 31^st-stage of each input matrix element; this corresponds to amaximum change in the last twelve bits; the statistical aspect is then in the choice of addition-subtraction. By comparing the results of the successive inversions, the magnitude of the probable error (as defined above) resulting from these perturbations can be rather closely determined. (These tests cover the case in which the errors of the input matrix elements are uncorrelated.) If the calculation is regarded as one in which the 30^th bit is the last significant one and the remaining 13 bits are guard digits, then, for a faithful algorithm, the observed probable error should appear starting in the 31^st place. This is found to be the case quite accurately for the algorithm here proposed, but not for other commonly used algorithms.It should be pointed out that the algorithm here described is optimal also for normalized arithmetic (in which errors are ignored), because the precision on the one hand cannot be increased by retaining insignificant digits and on the other hand is in any case limited by the true probable error, no matter what algorithm is used (unless the input data are infinitely precise). However, the results are likely to be less satisfactory in normalized arithmetic, because explicit use is made of the relative precisions of certain quantities at each pivoting in the matrix inversion, and the machine indication of this relative precision is falsified when numbers are unduly normalized.Work performed under the auspices of the U. S. Atomic Energy Commission.We are indebted to Prof. R. D.Richtmyer for his editorial comments that led to this abstract.     

Discrepancy in arithmetic progressions

It is proven that there is a two-coloring of the first integers for which all arithmetic progressions have discrepancy less than . This shows that a 1964 result of K. F. Roth is, up to constants, best possible.

     

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli and other small moduli.

     

Algorithms in unnormalized arithmetic

A class of recurrence relations is studied in the context of unnormalized arithmetic. The calculational structure of the proposed algorithm is based on error-theoretic considerations, and is designed to yield a measure of error in a natural manner. The algorithm gives consistent results whether the error propagation is stable or unstable.This work was supported by the Atomic Energy Commission on Contract AT (11-1)-614.     

Randomized proofs in arithmetic

A randomized proof system for arithmetic is introduced. A proof of an arithmetical formula is defined as its derivation from the axioms of arithmetic by the standard rules of inference of arithmetic and also one more rule which we call the random substitution rule. Such proofs can be regarded as a special kind of interactive proof and, more exactly, as a special kind of the Arthur-Merlin proofs. The main result of the paper shows that a proof in arithmetic with the random substitution rule can be considerably shorter than an arithmetical proof of the same formula. Namely, there exists a set of formulas such that (i) these formulas are provable in arithmetic but, unless PSPACE=NP, do not have polynomially long proofs; (ii) these proofs have polynomially long proofs in arithmetic with random substitution (whatever random numbers appear) and the probability of error of these proofs is exponentially small. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 49–71. Translated by E. Ya. Dantsin.