location:  Publications → journals
Search results

Search: MSC category 11D45 ( Counting solutions of Diophantine equations )

 Expand all        Collapse all Results 1 - 3 of 3

1. CMB 2011 (vol 56 pp. 258)

Chandoul, A.; Jellali, M.; Mkaouar, M.
 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 fieldsCategories:11A55, 11D45, 11D72, 11J61, 11J66

2. CMB 2008 (vol 51 pp. 337)

Bennett, Michael A.
 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)

McKinnon, David
 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
 top of page | contact us | privacy | site map |