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1. CMB 2011 (vol 56 pp. 258)
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 |
2. CMB 2008 (vol 51 pp. 337)
Differences between Perfect Powers We apply the hypergeometric method of Thue and Siegel to prove
that if $a$ and $b$ are positive integers, then the inequality $
0 <| a^x - b^y | < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$
has at most a single solution in positive integers $x$ and $y$.
This essentially sharpens a classic result of LeVeque.
Categories:11D61, 11D45 |
3. CMB 2004 (vol 47 pp. 264)
Counting Rational Points on Ruled Varieties In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.
Categories:11G50, 11D45, 11D04, 14G05 |