The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Igor E Shparlinski - One of the best experts on this subject based on the ideXlab platform.
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bounds of some double Exponential Sums
Journal of Number Theory, 2021Co-Authors: Nilanjan Bag, Igor E ShparlinskiAbstract:Abstract We give a new bound on double Sums with monomials over an interval and an arbitrary set in finite fields. This bound generalises and in some cases improves previous results.
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new estimates for Exponential Sums over multiplicative subgroups and intervals in prime fields
Journal of Number Theory, 2020Co-Authors: Daniel Di Benedetto, Igor E Shparlinski, M Z Garaev, Victor C Garcia, Diego Gonzalezsanchez, Carlos A TrujilloAbstract:Abstract Let H be a multiplicative subgroup of F p ⁎ of order H > p 1 / 4 . We show that max ( a , p ) = 1 | ∑ x ∈ H e p ( a x ) | ≤ H 1 − 31 / 2880 + o ( 1 ) , where e p ( z ) = exp ( 2 π i z / p ) , which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double Exponential Sums with product nx with x ∈ H and n ∈ N for a short interval N of consecutive integers.
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new estimates for Exponential Sums over multiplicative subgroups and intervals in prime fields
arXiv: Number Theory, 2020Co-Authors: Daniel Di Benedetto, Igor E Shparlinski, M Z Garaev, Victor C Garcia, Diego Gonzalezsanchez, Carlos A TrujilloAbstract:Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 \pi i z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double Exponential Sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.
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bounds of trilinear and trinomial Exponential Sums
arXiv: Combinatorics, 2020Co-Authors: Simon Macourt, Giorgis Petridis, Ilya D Shkredov, Igor E ShparlinskiAbstract:We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear Exponential Sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear Sums with all variables in multiplicative subgroups.
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bounds of trilinear and quadrilinear Exponential Sums
Journal D Analyse Mathematique, 2019Co-Authors: Giorgis Petridis, Igor E ShparlinskiAbstract:We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear Exponential Sums which improves the previous bound of Bourgain and Garaev (2009). We also obtain explicit bounds for quadrilinear Exponential Sums.
Richard J Lipton - One of the best experts on this subject based on the ideXlab platform.
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on tractable Exponential Sums
Lecture Notes in Computer Science, 2010Co-Authors: Jinyi Cai, Xi Chen, Richard J LiptonAbstract:We consider the problem of evaluating certain Exponential Sums. These Sums take the form Σx1,x2,...,xn∈ZN e2πi/N f(x1,x2,...,xn), where each xi is summed over a ring ZN, and f(x1,x2,...,xn) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree ≥ 3 we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem -- a complete classification of each problem to be either computable in polynomial time or #P-hard -- for a class of Exponential Sums. These Sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss Sums form the basic building blocks; for the hardness result we prove group-theoretic necessary conditions for tractability.
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on tractable Exponential Sums
arXiv: Computational Complexity, 2010Co-Authors: Jinyi Cai, Xi Chen, Richard J LiptonAbstract:We consider the problem of evaluating certain Exponential Sums. These Sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of Exponential Sums. These Sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss Sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N.
L V Meng - One of the best experts on this subject based on the ideXlab platform.
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incomplete Exponential Sums over galois rings and their applications in kerdock code sequences derived from z_p 2
Computer Science, 2013Co-Authors: L V MengAbstract:An upper bound for the incomplete Exponential Sums over Galois rings was derived.Based on the incomplete Exponential Sums,the nontrivial upper bound for the aperiodic autocorrelation of the Kerdock-code p-ary sequences derived from Zp2was given,where p is an odd prime.The result shows that these sequences have low aperiodic autocorrelation and provide strong potential applications in communication systems and cryptography.The estimate of the partial period distributions of these sequences was also derived.
Dengguo Feng - One of the best experts on this subject based on the ideXlab platform.
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incomplete Exponential Sums over galois rings with applications to some binary sequences derived from z sub 2 sup l
IEEE Transactions on Information Theory, 2006Co-Authors: Dengguo FengAbstract:An upper bound for the incomplete Exponential Sums over Galois rings is derived explicitly. Based on the incomplete Exponential Sums, we analyze the partial period properties of some binary sequences derived from Z/sub 2//sup l/ in detail, such as the Kerdock-code binary sequences and the highest level sequences of primitive sequences over Z/sub 2//sup l/. The results show that the partial period distributions and the partial period independent r-pattern distributions of these binary sequences are asymptotically uniform. Nontrivial upper bounds for the aperiodic autocorrelation of these sequences are also given.
Annemaria Ernvallhytonen - One of the best experts on this subject based on the ideXlab platform.
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on the error term in the approximate functional equation for Exponential Sums related to cusp forms
International Journal of Number Theory, 2008Co-Authors: Annemaria ErnvallhytonenAbstract:We give a proof for the approximate functional equation for Exponential Sums related to holomorphic cusp forms and derive an upper bound for the error term.
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on short Exponential Sums involving fourier coefficients of holomorphic cusp forms
International Mathematics Research Notices, 2008Co-Authors: Annemaria Ernvallhytonen, Kimmo KarppinenAbstract:We improve known estimates for the linear Exponential Sums containing Fourier coefficients of holomorphic cusp forms and show that in some cases, our bound is actually sharp. We also briefly visit nonlinear Exponential Sums, and prove some results concerning the density of rational numbers satisfying certain conditions.
