Integer relations among algebraic numbers

A vector m = ( m 1 , … , m n ) ∈ Z n ∖ { 0 } m = ({m_1}, \ldots ,{m_n}) \in {{\mathbf {Z}}^n}\backslash \{ 0\} is called an integer relation for the real numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , if ∑ α i m i = 0 \sum {\alpha _i}{m_i} = 0 holds. We present an algorithm that, when given algebraic numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} and a parameter ε \varepsilon , either finds an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} or proves that no relation of Euclidean length shorter than 1 / ε 1/\varepsilon exists. Each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation. Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs \[ poly ( log ⁡ 1 / ε , n , log ⁡ max i height ( α i ) , [ Q ( α 1 , … , α n ) : Q ] ) {\operatorname {poly}}\left ( {\log 1/\varepsilon ,n,\log \max \limits _i {\text {height}}({\alpha _i}),[{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}):{\mathbf {Q}}]} \right ) \] bit operations. The straightforward algorithm that works with a primitive element of the field extension Q ( α 1 , … , α n ) {\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}) of Q would take \[ poly ( n , log ⁡ max i height ( α i ) , ∏ i = 1 n degree ( α i ) ) {\operatorname {poly}}\left ( {n,\log \max \limits _i {\text {height}}({\alpha _i}),\prod \limits _{i = 1}^n {{\text {degree}}({\alpha _i})} } \right ) \] bit operations. In order to prove the correctness of the algorithm, we show a lower bound for | ∑ α 1 m i | \left | {\sum {\alpha _1}{m_i}} \right | if m is not an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , which may be interesting in its own right.

Let x = (x{sub 1}, x{sub 2} {hor_ellipsis}, x{sub n}) be a vector of real or complex numbers. x is said to possess an integer relation if there exist integers a{sub i}, not all zero, such that a{sub 1}x{sub 1} + a{sub 2}x{sub 2} + {hor_ellipsis} + a{sub n}x{sub n} = 0. By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we will see in the examples below, an integer relation algorithm can be used to recognize a computed constant in terms of a formula involving known constants, or to discover an underlying relation between quantities that can be computed to high precision. At the present time, the most effective algorithm for integer relation detection is the 'PSLQ' algorithm of mathematician-sculptor Helaman Ferguson [10, 4]. Some efficient 'multi-level' implementations of PSLQ, as well as a variant of PSLQ that is well-suited for highly parallel computer systems, are given in [4]. PSLQ constructs a sequence of integer-valued matrices B{sub n} that reduces the vector y = xB{sub n}, until either the relation is found (as one of the columns of B{sub n}), or else precision is exhausted. At the same time, PSLQ generates a steadily growing bound on the size of any possible relation. When a relation is found, the size of smallest entry of the vector y abruptly drops to roughly 'epsilon' (i.e. 10{sup -p}, where p is the number of digits of precision). The size of this drop can be viewed as a 'confidence level' that the relation is real and not merely a numerical artifact - a drop of 20 or more orders of magnitude almost always indicates a real relation. Very high precision arithmetic must be used in PSLQ. If one wishes to recover a relation of length n, with coefficients of maximum size d digits, then the input vector x must be specified to at least nd digits, and one must employ nd-digit floating-point arithmetic. Maple and Mathematica include multiple precision arithmetic facilities and Maple ships with a full implementation of PSLQ. One may also use any of several freeware multiprecision software packages, for example the ARPREC package by the first author and colleagues at LBNL [7]. In the remaining sections we describe various representative applications of PSLQ. More detail about these examples is given in [8] and the references therein.

The equal-tempered 10-tone scale e n/10 (n=0,±1,±2, …), using the Euler number e=2.71828… as a pseudo-octave is shown to approximate well the prime number harmonics 2, 3, 5, and 11. Equal-tempered scales simultaneously approximating certain frequency ratios, shall be called tonal scales.1 Some of the properties of the Euler scale and its relation to other tonal scales are explored. The general mathematical problem of identifying tonal scales can be solved by investigating integer relations, using the μEuclidean algorithm, a modification of the PSLQ algorithm. If restricted to two numbers, the μEuclidean algorithm goes over identically into the ancient Euclidean algorithm, contrary to the PSLQ algorithm. The μEuclidean is able to solve a certain class of higher dimensional integer relations where the PSLQ (not the PPSLQ) algorithm breaks down. In general, the μEuclidean algorithm finds smaller integer relations than the (P)PSLQ algorithm. In an appendix, a simple alternative procedure is presented for determining tonal scales based on continued fractions.

Applications of integer relation algorithms

On the hardness of approximating shortest integer relations among rational numbers

Let $\Gamma \subset \overline{\mathbb Q}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\linebreak\beta \in \overline{\mathbb Q}^\times $ be algebraic numbers with $\beta $ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in \Gamma \times \mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\delta qu|>1$ and $$\begin{align*} & 0<|\delta qu+\beta-p|<\frac{1}{H(u)^\varepsilon q^{d+\varepsilon}}, \end{align*}$$where $H(u)$ denotes the absolute Weil height. This is an inhomogeneous analogue of the main theorem in [ 2]. As an application of our result, we also prove a transcendence result, which states as follows: let $\alpha>1$ be a real number. Let $\beta $ be an algebraic irrational number and $\lambda $ be a non-zero real algebraic number. For a given real number $\varepsilon>0$, if there are infinitely many natural numbers $n$ for which $||\lambda \alpha ^n+\beta || < 2^{- \varepsilon n}$ holds true, then $\alpha $ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha $ and $\beta $ both are algebraic numbers satisfying same conditions, then a particular result of Kulkarni et al. [ 3] asserts that $\alpha ^d$ is a Pisot number. When $\beta $ is an algebraic irrational, our result implies that no algebraic number $\alpha $ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [ 7].

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

The Minimal Euclidean Norm of an Algebraic Number Is Effectively Computable

Let \(\alpha \) be an algebraic number of degree \(d\) with minimal polynomial \(F \in \mathbb {Z}[X]\), and let \(\mathbb {Z}[\alpha ]\) be the ring generated by \(\alpha \) over \(\mathbb {Z}\). We are interested whether a given number \(\beta \in \mathbb {Q}(\alpha )\) belongs to the ring \(\mathbb {Z}[\alpha ]\) or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number \(r/t \in \mathbb {Q}\), where \(r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1\), belongs to the ring \(\mathbb {Z}[\alpha ]\) if and only if the square-free part of its denominator \(t\) divides all the coefficients of the minimal polynomial \(F \in \mathbb {Z}[X]\) except for the constant coefficient \(F(0)\) that must be relatively prime to \(t\), namely \(\gcd (F(0),t)=1\). We also study the question when the equality \(\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']\) for algebraic numbers \(\alpha , \alpha '\) conjugates over \(\mathbb {Q}\) holds. In particular, it is shown that for each \(d \in \mathbb {N}\), there are conjugate algebraic numbers \(\alpha , \alpha '\) of degree \(d\) satisfying \(\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')\) and \(\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']\). The question concerning the equality \(\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']\) is answered completely for conjugate quadratic pairs \(\alpha ,\alpha '\) and also for conjugate pairs \(\alpha , \alpha '\) of cubic algebraic integers.

We present algorithms, which when given a real vector x∈ℝn and a parameter k∈ℕ as input either find an integer relation m∈ℤn, m≠0 with xTm=0 or prove there is no such integer relation with ‖m‖≦2k. One such algorithm halts after at most O(n3(k+n)) arithmetic operations using real numbers. It finds an integer relation that is no more than \(2^{\frac{{n - 2}}{2}}\)times longer than the length of the shortest relation for x. Given a rational input x∈ℚn this algorithm halts in polynomially many bit operations. The basic algorithm of this kind is due to Ferguson and Forcade (1979) and is closely related to the Lovasz (1982) lattice basis reduction algorithm.

We derive an algorithm based on the ellipsoid method that solves linear programs whose coefficients are real algebraic numbers. By defining the encoding size of an algebraic number to be the bit size of the coefficients of its minimal polynomial, we prove the algorithm runs in time polynomial in the dimension of the problem, the encoding size of the input coefficients, and the degree of any algebraic extension which contains the input coefficients. This bound holds even if all input and arithmetic is performed symbolically, using rational numbers only.

We present a new algorithm for reconstructing an exact algebraic number from its approximate value by using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on obtaining an exact rational number from its approximation. The algorithm is applicable for finding exact minimal polynomial of an algebraic number by its approximate root. This also enables us to provide an efficient method of converting the rational approximation representation to the minimal polynomial representation, and devise a simple algorithm to factor multivariate polynomials with rational coefficients.Compared with the subsistent methods, our method combines advantage of high efficiency in numerical computation, and exact, stable results in symbolic computation. The experimental results show that the method is more efficient than identify in Maple for obtaining an exact algebraic number from its approximation. Moreover, the Digits of our algorithm is far less than the LLL-lattice basis reduction technique in theory. In this paper, we completely implement how to obtain exact results by numerical approximate computations.

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number e, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+e, where H(α) denotes the naive height of α. We sharpen this result by replacing e by a function H 7→ e(H) that tends to zero when H tends to infinity. We make a similar improvement for the approximation to algebraic numbers by algebraic integers, as well as for an inhomogeneous approximation problem.

Based on an improved parameterized integer relation construction method, a complete algorithm is proposed for finding an exact minimal polynomial from its approximate root. It relies on a study of the error controlling for its approximation. We provide a sufficient condition on the precision of the approximation, depending only on the degree and the height of its minimal polynomial. Our result is superior to the existent error controlling on obtaining an exact rational or algebraic number from its approximation. Moreover, some applications are presented and compared with the subsistent methods.

Comments on the height reducing property II