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Search: MSC category 11T23 ( Exponential sums )

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1. CJM Online first

Macourt, Simon; Shkredov, Ilya D.; Shparlinski, Igor E.
 Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields We give a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials. Keywords:exponential sum, sparse polynomial, trinomialCategories:11L07, 11T23

2. CJM Online first

Ha, Junsoo
 Smooth Polynomial Solutions to a Ternary Additive Equation Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite field of $q$ elements, and $Y$ be a large integer. We say a polynomial in $\mathbf{F}_{q}[T]$ is $Y$-smooth if all of its irreducible factors are of degree at most $Y$. We show that a ternary additive equation $a+b=c$ over $Y$-smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and $s$ is large, we prove that the $S$-unit equation $u+v=1$ has at least $\exp(s^{1/6-\epsilon}\log q)$ solutions. Keywords:smooth number, polynomial over a finite field, circle methodCategories:11T55, 11D04, 11L07, 11T23

3. CJM 2005 (vol 57 pp. 338)

Lange, Tanja; Shparlinski, Igor E.
 Certain Exponential Sums and Random Walks on Elliptic Curves For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=$$z_1, \dots, z_T$$$ such that $k z_1,\dots, kz_T$ is a permutation of $z_1, \dots, z_T$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$. We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings. Categories:11L07, 11T23, 11T71, 14H52, 94A60

4. CJM 2003 (vol 55 pp. 225)

Banks, William D.; Harcharras, Asma; Shparlinski, Igor E.
 Short Kloosterman Sums for Polynomials over Finite Fields We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring $\mathbb{F}_q[x]/M(x)$ for collections of polynomials either of the form $f^{-1}g^{-1}$ or of the form $f^{-1}g^{-1}+afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields. Categories:11T23, 11T06
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