Linear forms in the logarithms of algebraic numbers

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$ , a square matrix $A\in \mathbb {Q}^{d\times d}$ , and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$ , and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Anx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

On linear forms in the logarithms of algebraic numbers

Linear forms in the logarithms of algebraic numbers (II)

On a Family of Quintic Thue Equations

On the number of solutions of binomial Thue inequalities

Pillai's conjecture revisited

This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker’s theory of linear forms in logarithms of algebraic numbers, along with a variation of the Baker–Davenport reduction method, to solve the Diophantine equation [Formula: see text], where [Formula: see text], and [Formula: see text] are non-negative integers, as presented here. Furthermore, the maple[Formula: see text] codes used in the computations made throughout the paper are also shared.

We begin with some basic tools necessary for the proof of Theorem 7.1.1 in Sect. 8.1. First, Theorem 7.1.1 is reduced to an equivalent statement; see Theorem 8.1.2. In Sect. 8.1.1, we derive a simple, but useful, non-trivial lower bound for a non-vanishing linear form in logarithms of algebraic numbers with bounded coefficients. Section 8.1.2 provides construction of an augmentative polynomial. In Sect. 8.1.3, we give the construction of the auxiliary polynomial \(\Phi (Z_0,\ldots ,Z_{n-1})\) in several variables which generalises the function of a single complex variable employed by Gelfond. Basic estimates on \(\Phi \) are shown in Sect. 8.1.4. The main difficulty is in the interpolation techniques. Usually the order of the derivatives is increased while leaving the points of interpolation fixed. Baker used a special extrapolation procedure in which the range of interpolation points is extended while the order of the derivatives is reduced, and the absolute values of these derivatives are shown to be very small. See Sects. 8.1.5 and 8.1.6.

In this automata column, Jöel Ouaknine and James Worrell beautifully write about beautiful recent work on linear recurrence sequences, which are sequences of integers defined by recursion like in the Fibonacci sequence. Clearly such sequences can appear almost everywhere, so there is no particularly good reason to talk about them in the automata column, but the authors politely point out such sequences can be used to count the number of words of given length accepted by an automaton. Thanks to this connection we get to have an automata column that cites Terrence Tao and uses logarithms of algebraic numbers.

The equations considered in this chapter are in essence different both from the Thue equation and from its direct generalisations. It is possible to prove the existence of an effective bound for solutions of these equations by purely arithmetic methods, by reducing them to the Thue equation or to its generalisations over relative fields. However the bounds obtained in this way are not quite satisfactory in the general case, and we again turn to exponential equations to obtain better results. We give an analysis of the hyperelliptic equation and of integer and S-integer points on elliptic curves.

On differences of perfect powers and prime powers

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q. How to deal with standard Weierstras equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10

We apply the methods and considerations described in the previous chapter to more general situations so as to obtain new facts generalising our former results. Then we proceed to a new type of equations in which at least one of the unknowns is a power of an unknown integer. Our aim is to bound the unknown exponent so as to reduce these new equations to those of the kind considered before. In this way we determine, for example, that any integral polynomial having at least two simple roots represents only a finite number of powers of integers with exponents greater than 2. We also give an analysis of S-integer solutions of the Catalan equation. *** DIRECT SUPPORT *** A00I6B17 00003