1. CJM Online first
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 method Categories: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 BrunTitchmarsh theorem for
polynomials over finite fields.
Categories:11T23, 11T06 
