Results 1  10
of
35
Closed forms: what they are and why we care
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not cl ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.
INTEGER SYMMETRIC MATRICES OF SMALL SPECTRAL RADIUS AND SMALL MAHLER MEASURE
, 907
"... Abstract. In a previous paper we completely described cyclotomic matrices—integer symmetric matrices of spectral radius at most 2. In this paper we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In a previous paper we completely described cyclotomic matrices—integer symmetric matrices of spectral radius at most 2. In this paper we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular we solve the strong version of Lehmer’s problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least ‘Lehmer’s number ’ λ0 = 1.17628.... 1. Statement of results For a monic polynomial g(x) with integer coefficients and degree d, define z d g(z+1/z), a monic reciprocal polynomial of degree 2d, to be its associated reciprocal polynomial. If g(z) has all its roots real and in the interval [−2, 2], then the roots of its associated reciprocal polynomial are all of modulus 1, and, by a theorem of Kronecker [13], it is a cyclotomic polynomial. If A is a dbyd symmetric matrix with integer entries, then all the roots of its characteristic polynomial are real algebraic integers, and we denote by RA(z) its associated reciprocal polynomial. If A has spectral radius at most 2, so that RA(z) is cyclotomic, then
DISTRIBUTION OF ALGEBRAIC NUMBERS
"... Abstract. Schur studied limits of the arithmetic means An of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that lim sup n→ ∞ An  ≤ 1 − √ e/2. We show that An → 0, and estimate the rate o ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Schur studied limits of the arithmetic means An of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that lim sup n→ ∞ An  ≤ 1 − √ e/2. We show that An → 0, and estimate the rate of convergence by generalizing the ErdősTurán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for lim infn→ ∞ An was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure. 1. Schur’s problems on means of algebraic numbers Let E be a subset of the complex plane C. Consider the set of polynomials Zn(E) of the exact degree n with integer coefficients and all zeros in E. We denote a subset of Zn(E) with simple zeros by Zs n(E). Given M> 0, we write Pn = anzn +... ∈ Zs n(E, M) if an  ≤ M and Pn ∈ Zs n(E) (respectively Pn ∈ Zn(E, M) if an  ≤ M and Pn ∈ Zn(E)). Schur [46, §48] studied the limit behavior of the arithmetic means of zeros for polynomials from Zs n(E, M) as n → ∞, where M> 0 is an arbitrary fixed number. His results may be summarized in the following statements. Let R+: = [0, ∞), where R is the real line. ∏ n k=1 Theorem A (Schur [46, p. 393], Satz IX) Given a polynomial Pn(z) = an (z − αk,n), define the arithmetic mean of squares of its zeros by Sn: = ∑n k=1 α2 k,n /n. If Pn ∈ Zs n(R, M) is any sequence of polynomials with degrees n → ∞, then
Topological entropy and algebraic entropy for group endomorphisms
 Proceedings ICTA2011 Islamabad, Pakistan July 410 2011 Cambridge Scientific Publishers
, 2012
"... ar ..."
(Show Context)
Minimal Mahler measures
, 2007
"... We determine the minimal Mahler measure of a primitive, irreducible, noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive, irreducible, noncyclotomic polynomials with measure less than 1.3 and degree at most 44. 1 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We determine the minimal Mahler measure of a primitive, irreducible, noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive, irreducible, noncyclotomic polynomials with measure less than 1.3 and degree at most 44. 1
Reconstructing Numbers from Pairwise Function Values
"... Abstract. The turnpike problem is one of the few “natural ” problems that are neither known to be NPcomplete nor solvable by efficient algorithms. We seek to study this problem in a more general setting. We consider the generalized problem which tries to resolve set A = {a1,a2, ·· ·,an} from pairwi ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The turnpike problem is one of the few “natural ” problems that are neither known to be NPcomplete nor solvable by efficient algorithms. We seek to study this problem in a more general setting. We consider the generalized problem which tries to resolve set A = {a1,a2, ·· ·,an} from pairwise function values {f(ai,aj)1 ≤ i, j ≤ n} for a given bivariate function f. WecallthisproblemtheNumber Reconstruction problem. Our results include efficient algorithms when f is monotone and nontrivial bounds on the number of solutions when f is the sum. We also generalize previous backtracking and algebraic algorithms for the turnpike problem such that they work for the family of antimonotone functions and lineardecomposable functions. Finally, we propose an efficient algorithm for the string reconstruction problem, which is related to an approach to protein reconstruction. 1
NEGATIVE PISOT AND SALEM NUMBERS AS ROOTS OF NEWMAN POLYNOMIALS
"... Abstract. A Newman polynomial has all its coefficients in {0, 1} and constant term 1. It is known that every root of a Newman polynomial lies in the slit annulus {z ∈ C: τ −1 < z  < τ} \ R +, where τ denotes the golden ratio, but not every polynomial having all of its conjugates in this set ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. A Newman polynomial has all its coefficients in {0, 1} and constant term 1. It is known that every root of a Newman polynomial lies in the slit annulus {z ∈ C: τ −1 < z  < τ} \ R +, where τ denotes the golden ratio, but not every polynomial having all of its conjugates in this set divides a Newman polynomial. We show that every negative Pisot number in (−τ, −1) with no positive conjugates, and every negative Salem number in the same range obtained by using Salem’s construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also construct a number of polynomials having all their conjugates in this slit annulus, but that do not divide any Newman polynomial. Finally, we determine all negative Salem numbers in (−τ, −1) with degree at most 20, and verify that every one of these is satisfied by a Newman polynomial. 1.
HOLOMORPHIC FUNCTIONS ON BUNDLES OVER ANNULI
, 2008
"... ABSTRACT. We consider a family ˘ Em(D, M) ¯ of holomorphic bundles constructed as follows: to any given M ∈ GLn(Z), we associate a “multiplicative automorphism ” ϕ of (C ∗ ) n. Now let D ⊆ (C ∗ ) n be a ϕinvariant Stein Reinhardt domain. Then Em(D, M) is defined as the flat bundle over the annulus ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We consider a family ˘ Em(D, M) ¯ of holomorphic bundles constructed as follows: to any given M ∈ GLn(Z), we associate a “multiplicative automorphism ” ϕ of (C ∗ ) n. Now let D ⊆ (C ∗ ) n be a ϕinvariant Stein Reinhardt domain. Then Em(D, M) is defined as the flat bundle over the annulus of modulus m> 0, with fiber D, and monodromy ϕ. We show that the function theory on Em(D, M) depends nontrivially on the parameters m,M and D. Our main result is that Em(D, M) is Stein if and only if mlog ρ(M) ≤ 2π 2, where ρ(M) denotes the max of the spectral radii of M and M −1. As corollaries, we: – obtain a classification result for Reinhardt domains in all dimensions; – establish a similarity between two known counterexamples to a question of J.P. Serre; – suggest a potential reformulation of a disproved conjecture of Siu Y.T. Let D be a Stein manifold. We say that D belongs to S when: for any Stein manifold B and any locally trivial bundle E → B, the manifold E is also Stein. A famous question of Serre can be formulated as: “Are all manifolds in S?” Skoda answered it in the negative, by proving that C 2 ̸ ∈ S (cf. [Sko]). Mok showed that any open Riemann surface belongs to S (cf. [Mok]). Many bounded domains belong to S: for any given bounded domain D, DiederichFornæssStehlé showed that if ∂D is smooth, then D ∈ S (cf. [DieFor] and [Steh]). Siu showed that if b1(D) = 0 then D ∈ S (cf. [Siu]). However, Cœuré and Lœb (cf. [CœLœ]) found a bounded domain DCL not in S. It happens that DCL has the Reinhardt symmetry. It is an open problem to characterize bounded domains of C d not in S (cf. [ChZh]). This classification problem is solved for all bounded Reinhardt domains with d = 2 in [PflZwo] and with d = 2 or 3 in [OelZaf]. Here we study holomorphic functions on a family of bundles {Em(D,M)} over annuli, depending on a non necessarily bounded Reinhardt domain