The p ‐adic generalization of the Thue‐Siegel‐Roth theorem

with integral coefficients from the field of rational numbers or from some quadratic field. The quantity A is defined for convenient reference. We can take ap, O without any loss of generality, by the use (if necessary) of linear transformations of determinant unity (so that the number of integral solutions is not changed). First, suppose that the coefficients of (2) are rational integers. If A is negative, then the graph of (2) is finite in extent, and there is at most a finite number of solutions in integers. If A >0, the graph of (2) is a parabola, an hyperbola, or two straight lines, and we prove the following result.

Under general assumptions on the functions and it is proved that the inequality where is the distance from to the nearest integer and , , has only a finite number of solutions in integers for almost all . This establishes the extremality of the surface .Bibliography: 11 titles.

An algorithm for the biobjective integer minimum cost flow problem

—Schmidt5 proved that for almost all real numbers α, the number of solutions in integers p, q of the inequalities $$ \left| {q\alpha - p} \right| < 1/qand1\underline \leqslant q\underline \leqslant B$$ is asymptotic to a constant times log B. One might conjecture that the classical numbers (e.g., algebraic numbers, e, π) behave like almost all numbers. Machine computations’ were carried out for some of these numbers, and they seemed to bear out such a conjecture. Also, Lang3 has proved that the estimate is valid when α is a real quadratic irrationality.

Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

Let θ 1 , …, θ k be k real numbers. Suppose ψ( t ) is a positive decreasing function of the positive variable t . Define λ( N ), for all positive integers N , to be the number of solutions in integers p 1 …, p k , q of the inequalities and

We are interested in the following problem: Given two (distinct) real algebraic numbers in isolating interval representation, that is an isolating interval with rational endpoints and a square free polynomial with integer coefficients, can we compute a number between them as a rational function of the coefficients of the polynomials that define these two numbers? Assume that the order of the two numbers is known (we will remove this assumption in the sequel). If we are given intervals that contain the real algebraic numbers and a procedure to refine them, we can solve our problem as follows: We refine the intervals until they become disjoint, this will happen eventually since we assume that the algebraic numbers are not equal, and then we compute a rational between the intervals, which separates the algebraic numbers. However, this iterative approach depends on separation bounds, e.g. [7]. We present a direct method, which is applicable when we allow in addition to compute the floor of a polynomial expression that involves real algebraic numbers. The problem arises when we wish to compute rational numbers that isolate the roots of an integer polynomial of small degree, say ≤ 5 [2]. Also in geometry, in order to analyse the intersection of two quadrics P and Q [6], one needs to determine the real roots of the polynomial det( P + xQ ) = 0, their multiplicities and a value in between each of these roots. Another motivation comes from the arrangement of quadrics [4] In this case a rational is needed separating two real roots of two polynomials with real algebraic numbers as coefficients. The real roots of such polynomials can be expressed as real algerbaic numbers and so we face the problem of computing a rational separating two real algebraic numbers.

Real algebraic numbers are the real numbers that are real roots of univariate polynomials with integer coefficients. We study exact algorithms, from a theoretical and an implementation point of view, based on integer arithmetic of arbitrary precision, for computations with real algebraic numbers and applications of these algorithms on problems and algorithms in non linear computational geometry. In order to construct a real algebraic number we must compute the real roots of a univariate polynomial with integer cofficients. We unify and simplify the theory behind the subdivision based algorithms for real root isolation and we improve the complexity of the algorithm that is based on the continued fraction expansion of the real numbers. The best known complexity bound up today is achieved using new techniques. Moreover, we prove that the bound holds for non square-free polynomials and that in the same complexity bound we can compute the multiplicities of the real roots. We prove a new bound for the expected complexity of the algorithm based on continued fractions. We generalize the real root isolation algorithms to bivariate polynomial systems. Our experimental analysis proves the effectiveness of our methods. The algorithms that we consider for computations with real algebraic numbers are construction, comparison, sign evaluation and quantifier elimination. If the degree of the polynomial is small, i.e. ≤ 4 in the univariate case and ≤ 2 in the bivariate case, we propose special purpose algorithms that have constant arithmetic complexity. For all the algorithms we present a C++ implementation and an experimental analysis. In computational geometry we study the predicates needed by the algorithms for the arrangement of elliptic arcs in the plane and the computation of the Voronoi diagram of ellipses, also in the plane. Finally, given a convex lattice polygon we study algorithms for decomposing it to two other convex lattice polygons, such that their Minkowski sum is the original polygon.

Does Computer Algebra help at all learning about real numbers?

For any positive integer [Formula: see text], we state and prove a formula for the number of solutions in integers of [Formula: see text]

4 - Linear Equations with Unit Coefficients

has no integer solutions with X,Y, Z ≥ 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation has only a finite number of solutions (for each given n, up to rescaling). This paper is an appreciation of some of the topological intuitions behind number theory. It aims to trace a logical path from the classification of surface diffeomorphisms to the proof of Finite Fermat. The route we take is the following.

Approximation par des nombres algébriques

has infinitely many solutions in algebraic numbers α of degree at most n. Here H(α) denotes the height of α, that is the maximum of the absolute values of the coefficients of the minimal polynomial of α. In 1965, Sprindzhuk proved that kn(ξ) = n + 1 for almost all ξ ∈ R (with respect to the Lebesgue measure on R) and kn(ξ) = n+1 2 for almost all ξ ∈ C (with respect to the Lebesgue measure in C). In 1971 Schmidt proved that if ξ is a real algebraic number of degree d, then kn(ξ) = min(d, n+ 1). So as for approximation by algebraic numbers of degree at most n, real algebraic numbers ξ of degree larger than n show the same behavious as almost all real numbers. Up to now, nobody had computed kn(ξ) for complex algebraic numbers ξ. I will present some new results in this direction, obtained jointly with Yann Bugeaud. These results show that if ξ is complex algebraic of degree d then kn(ξ) = d2 if d ≤ n and kn(ξ) ∈ { n+1 2 , n+2 2 } if d > n. The hard core of the proof of this result is a central result in Diophantine approximation, Schmidt’s Subspace Theorem.