arXiv subjects
Publications and source records attributed to Igor E. Shparlinski.
Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer, has order of magnitude $1/°g$.
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. Some of the families we consider are much thinner than the ones previously studied.
Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are infinitely many members of ${\mathbb P}^c$ having at most $R(c)$ prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and also when $c$ is large.
We improve several recent results by Hong, Lee, Lee and Park (2012) on gaps and Bzdȩga (2014) on jumps amongst the coefficients of cyclotomic polynomials. Besides direct improvements, we also introduce several new techniques that have never been used in this area.
We obtain optimal lower bounds for moments of theta functions. On the other hand, we also get new upper bounds on individual theta values and moments of theta functions on average over primes. The upper bounds are based on bounds of character sums and in particular on a modification of some recent results of M. Z. Garaev.
Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if $f(u) = v$. We obtain some theoretic estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity increases only slightly, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree $d$ over ${\mathbb F}_q$. Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\left(ζ(6) \prod_{k=4}^n ζ(k)\right)^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.
We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.
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 $\bigl\{s(m,n):m\in(\mathbb Z/n \mathbb Z)^*\bigr\}$ on average for $n\in[1,N]$.
We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V α_u β_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X] $ and some complex weights $\{α_u\}$, $\{β_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of ${\mathbb F}_p$.
For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes p there are enough small Elkies primes l to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1)) expected time.
We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.
We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences $$ \sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p, $$ with variables from rather general sets.
We obtain a tight, up to a logarithmic factor, upper bound on the number of solutions to the equation $$ \sum_{j=1}^n a_j \frac{s_j}{r_j} =a_0, \qquad $$ with variables $r_1,...,r_n$ in an arbitrary box at the origin and variables $s_1,..., s_n$ in an essentially arbitrary translation of this box. We apply this result to get an upper bound on the number of stochastic matrices with rational entries of bounded height.
Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is divisible by m. The running time of the algorithm is mp^(1/2+o(1)), and this leads to more efficient constructions of rational functions over F_p whose image is small relative to p. We also give an unconditional version of the algorithm that works for almost all primes p, and give a probabilistic algorithm with subexponential time complexity.
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 $p$.
We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from coefficients of several $L$-functions of elliptic curves and modular forms. In particular, we show that $|τ(n)|\le n^{11/2} (\log n)^{-1/2+o(1)}$ for a set of $n$ of asymptotic density 1, where $τ(n)$ is the Ramanujan $τ$ function while the standard argument yields $\log 2$ instead of $-1/2$ in the power of the logarithm. Another consequence of our result is that in the number of representations of $n$ by a binary quadratic form one has slightly more than square-root cancellations for almost all integers $n$. In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.
We use bounds of mixed character sums modulo a square-free integer $q$ of a special structure 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 $f_i \in {\mathbb Z}[X]$ and nonzero integers $a$ and $k_i$, $i=1, \ldots, n$. 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 hypersurface is known as the Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \ldots = f_n(X) = X^n\quad \text{and} \quad k_1 = \ldots = k_n =1 $$ it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.