Nonzero Integer Research Articles

Thenth root of a braid is unique up to conjugacy

We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.

Zeros of the Alternating Zeta Function on the Line ℝ(s) = 1

'*(s) = (1 21-s)'(s), (1) which is easily established, first for 91(s) > 1 by combining terms in the convergent Dirichlet series, and then by using analytic continuation to extend the result to the entire s-plane (see Theorem 13.11 in [1]). The factor (1 21-s) has a simple zero at s = 1 that cancels the pole of a(s), so (1) shows that <*(s) is an entire function of s. It vanishes at each zero of the factor (1 21-s) with the exception of s = 1, at which point *(1) = log 2. The other zeros of the factor (1 21-s) lie on the line 9i(s) = 1 and occur at the points s = 1 + 2kr i/ log 2 for all nonzero integers k. This note deduces the vanishing of *(s) at the zeros of (1 21-s), except for s = 1, without using (1). Instead, we use identity (4) relating the partial sums

Closed model categories for uniquely S-divisible spaces

For each integer n>1 and a multiplicative system S of non-zero integers, we give a distinct closed model category structure to the category of pointed spaces Top ★ and we prove that the corresponding localized category Ho(Top ★ ( S, n) ), obtained by inverting the weak equivalences, is equivalent to the standard homotopy category of uniquely ( S, n)-divisible, ( n−1)-connected spaces. A space X is said to be uniquely ( S, n)-divisible if for k⩾ n the homotopy group π k X is uniquely S-divisible. This equivalence of categories is given by an ( S, n)-colocalization functor that carries a pointed space X to a space X ( S, n) . There is also a natural map X ( S, n) → X which is (finally) universal among all the maps Z→ X with Z a uniquely ( S, n)-divisible, ( n−1)-connected space. The structure of closed model category given by Quillen to Top ★ is based on maps which induce isomorphisms on all homotopy group functors π k and for any choice of base point. For each pair ( S, n), the closed model category structure given here take as weak equivalences those maps that for the given base point induce isomorphisms on the homotopy groups functors π k( Z[S −1];−) with coefficients in Z[S −1] for k⩾ n. We note that the category Ho( Top ★ ( Z−{0},2) ) is the homotopy category of rational 1-connected spaces.

BOUNDS ON EXPONENTIAL SUMS AND THE POLYNOMIAL WARING PROBLEM MOD p

TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE Abstract. We give estimates for the exponential sum ∑p x=1 exp(2πif(x)/p), p a prime and f a non-zero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod p; namely the smallest γ such that f(x1) + · · ·+ f(xγ) ≡ N (mod p) is solvable in integers for any N .

On the enumerative geometry of real algebraic curves having many real branches

Let C be a smooth real plane curve. Let c be its degree and g its genus. We assume that C has at least g real branches. Let d be a nonzero natural integer strictly less than c. Let e be a partition of cd of length g. Let n be the number of all real plane curves of degree d that are tangent to g real branches of C with orders of tangency e1; .. .; eg. We show that n is finite and we determine n explicitly.

On the Diophantine Equations ${n\choose 2}=cx^l$ and ${n\choose 3}=cx^l$

Upper bounds for the solution $l$ of the equations of the title are derived by using results concerning the equation $ax^l-by^l=c$ with $a,b,c$ non-zero integers. These solutions are also determined in some special cases.

A lower bound for certain Ramsey type problems

One of the main results in this article is to present a new and simple proof of the following theorem; The Diophantine equation a 1/y 1 + a 2/y 2 + ··· + an /y n = 0, where yi ≠ 0 are n unknown variables and ai 's are non-zero integers, is regular if and only if there exists a non empty subset I of {1, 2, ···, n} such that ∑ i∈I ai = 0. In the rest of this article, we study a particular case when n = 3 in the above equation. Also, we define f(r) to be the largest integer such that if the integers in [1,f(r)] are r coloured, then no monochromatic solution to the Diophantine equation 1/x + 1/y = 1/z, x, y, z can exist. Then we prove that f(r) ≥ 2 r-3.192 - 1 and f(r + 2) ≥ 4 (f(r) + √(f(r)/5).

Polarizations on abelian varieties

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.

Small Prime Solutions of Quadratic Equations

AbstractLet b1,…,b5 be non-zero integers and n any integer. Suppose that b1 + … + b5 ≡ n (mod 24) and (bi, bj) = 1 for 1 ≤ i &lt; j ≤ 5. In this paper we prove that(i)if all bj are positive and , then the quadratic equation is soluble in primes pj, and(ii)if bj are not all of the same sign, then the above quadratic equation has prime solutions satisfying .

On the size of Diophantine m-tuples

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation

In this paper, we refine work of Beukers, applying results from the theory of Pade approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to \(\sqrt m\) (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have $$\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}$$ provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.

Effective solution of two simultaneous Pell equations by the elliptic logarithm method

where A1, A2 are non-zero integers, D1, D2 are non-square positive integers and A1D2−A2D1 6= 0. Note that, if the last condition is not satisfied, then A2/A1 = D2/D1 = q 2 for some q ∈ Q and the solutions (X, Y, Z) of the system of equations are given by (X, Y, qX), where (X, Y ) is a solution of the first equation of the system. A large number of papers is devoted to the study of simultaneous Pell equations; they could be classified into three categories: The first one includes papers, in which results are proved of the type “there are no solutions”, or “there are at most k solutions”, where k is a number between, say, 1 and 4; such are the papers [4],[5],[6],[7],[8],[20],[21], [26],[38],[39],[40]. The generality of such results is obtained at the cost of restricting at least two parameters among A1, D1, A2, D2. Of course, if a system (1) falls within the scope of such a paper and 3 one is so lucky that ∗Department of Mathematics, University of Crete, Iraklion, Greece, e-mail: tzanakis@math.uoc.gr , http://www.math.uoc.gr/ tzanakis Throughout this paper the term “algorithm”, in a broad sense though, could also be used in place of the word “method”. Often, solving (1) is reduced to finding common terms of two distinct second order recurrence sequences, a problem studied in a number of papers without any direct reference to (1); in our bibliography it is natural to include such papers. This is the very optimistic case!

On the number of real hypersurfaces hypertangent to a given real space curve

Let $C$ be a smooth geometrically integral real algebraic curve in projective $n$-space $\mathbb{P}^n$. Let $c$ be its degree and let $g$ be its genus. Let $d$, $s$ and $m$ be nonzero natural integers. Let $\nu$ be the number of real hypersurfaces of degree $d$ that are tangent to at least $s$ real branches of $C$ with order of tangency at least $m$. We show that $\nu$ is finite if $s=g$, $gm=cd$ and the restriction map $H^0(\mathbb{P}^n,\mathcal{O}(d))\rightarrow H^0(C,\mathcal{O}(d))$ is an isomorphism. Moreover, we determine explicitly the value of $\nu$ in that case.

On Representations of Integers by Indefinite Ternary Quadratic Forms

Let f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that −qdet(f) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f(x)=q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T→∞. We deduce from the results of our joint paper with Z. Rudnick that N(T, f, q)∼cEHL(T, f, q) as T→∞, where EHL(T, f, q) is the Hardy–Littlewood expectation (the product of local densities) and 0⩽c⩽2. We give examples of f and q such that c takes the values 0, 1, 2.

On Some Exponential Equations of S. S. Pillai

AbstractIn this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ax – by = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Rank-one power weakly mixing non-singular transformations

We show that Chacon's non-singular type III _\lambda transformation T_\lambda , 0 , is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\} , T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.

On the Homflypt skein module of $S^1 \times S^2$

Let k be a subring of the field of rational functions in x, v,s which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$ . If M is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$ ; (2) L with a positiv e twist $=(xv^{-1})L$ ; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of $S(S^1\times D^2)$ given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for $S(S^1\times D^2)$ based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements $s^{2n}-1$ , for n a nonzero integer, and the elements $s^{2m}-v^{2}$ , for any integer m, are invertible ink, then $S(S^{1} \times S^2)=k$ -torsion module $\oplus k$ . Here the free part is generated by the empty link $\phi$ . In addition, if the elements $s^{2m}-v^{4}$ , for m an integer, are invertible in k, then $S(S^{1} \times S^2)$ has no torsion. We also obtain some results for more general k.

Diophantine Equations for Second-Order Recursive Sequences of Polynomials

Let B be a nonzero integer. Let define the sequence of polynomials Gn(x) by G0(x) = 0, G1(x) = 1, Gn+1(x) = xGn(x) +BGn−1(x), n ∈ N. We prove that the diophantine equation Gm(x) = Gn(y) for m,n ≥ 3, m 6= n has only finitely many solutions.

Applications et morphismes harmoniques à valeurs dans une surface

In this paper, we introduce notions of Hermitian geometry to construct harmonic morphisms to surfaces. As an application, we prove that the standard conformal class of S 4 contains a family of metrics {g k,ℓ} parametrized by pairs of positive non-zero integers (k,ℓ) such that, to each pair, we can associate a harmonic morphism from ( S 4,g k,ℓ) to the Riemann sphere S 2 .

Generating Uniform Random Vectors

In this paper, we study the rate of convergence of the Markov chain Xn+1=AXn+bn mod p, where A is an integer matrix with nonzero integer eigenvalues and {bn}n is a sequence of independent and identically distributed integer vectors. If λi≠±1 for all eigenvalues λi of A, then n=O((log p)2) steps are sufficient and n=O(log p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalue λ1=±1, and λi≠±1 for all i≠1, n=O(p2) steps are necessary and sufficient.