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Discovery Projects - Grant ID: DP130100237

Title
Discovery Projects - Grant ID: DP130100237
Funding
ARC | Discovery Projects
Contract (GA) number
DP130100237
Start Date
2013/01/01
End Date
2015/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP130100237

 

  • On the Greatest Common Divisor of Shifted Sets

    Heyman, Randell; Shparlinski, Igor E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.

    Exponential Sums over Points of Elliptic Curves

    Ahmadi, Omran; Shparlinski, Igor E. (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.

    On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces

    We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some polynomials $f_i \in {\mathbb Z}[X]$, nonzero integer $a$ and positive integers $k_i$ $i=1, \ldots, n$. In the case of $$ f_1(X) = \ldots = f_n(X) = X^2 \quad \text{and}\quad k_1 = \ldots = k_n =1 $$ the above congruence is known as the Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \ldots = f_n(X)...

    Fractional parts of Dedekind sums

    Banks, William D.; Shparlinski, Igor E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec~(1997) on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums $s(m,n)$ with $m$ and $n$ running over rather general sets. Our result extends earlier work of Myerson (1988) and Vardi (1987). Using different techniques, we also study the least denominator of the collection of Dedekind sums $\bigl\{s(m,n):m\in(\mat...

    On the lattices from elliptic curves over finite fields

    In this paper, we continue the recent work of Fukshansky and Maharaj on lattices from elliptic curves over finite fields. We show that there exist bases formed by minimal vectors for these lattices except only one case. We also compute their determinants, and obtain sharp bounds for the covering radius.

    Explicit form of Cassels' $p$-adic embedding theorem for number fields

    Dubickas, Arturas; Sha, Min; Shparlinski, Igor E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

    On the singularity of the Demjanenko matrix of quotients of Fermat curves

    Fite, Francesc; Shparlinski, Igor E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) ...

    Additive Decompositions of Subgroups of Finite Fields

    We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalisations of results of C. Dartyge and A. Sarkozy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such the Karatsuba bound of double character sums and some results from additive combinatorics.

    Averaging operators over homogeneous varieties over finite fields

    Koh, Doowon; Shen, Chun-Yen; Shparlinski, Igor (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersectio...

    Lang-Trotter and Sato-Tate Distributions in Single and Double Parametric Families of Elliptic Curves

    Sha, Min; Shparlinski, Igor E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100237 (DP130100237)
    We obtain new results concerning Lang-Trotter conjecture on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A.C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....
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