Binary polynomial power sums vanishing at roots of unity

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$. We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$

Computing integral points on X_ns^+(p)

We describe an algorithm for computing integral points on the modular curve of prime level p associated to the normalizer of a non-split Cartan subgroup of GL_2(F_p). Using our method, we show that for 7<p<101 the only integral points on this curve are the CM-points

Elliptic curves over finite fields with Fibonacci numbers of points

For a prime power q and an elliptic curve E over Fq having q+1-a points, where a ∈ [-2√q,2√q], let #Em, m ≥ 1, be the sequence of numbers whose mth term is the number of points of E over Fqm. In this paper, we determine all instances when #({#Em}∩{F_n})≥ 2, where {Fn} (n≥1) is the sequence of Fibonacci numbers. That is, we determine all six--tuples (a,q,m1,m2,n1,n2) such that #E=q+1-a, #Em1=Fn1 and #Em2=Fn2

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

Back to the "Gold Standard": How Precise is Hematocrit Detection Today?

Introduction: The commonly used method for hematocrit detection, by visual examination of microcapillary tube, known as "micro-HCT", is subjective but remains one of the key sources for fast hematocrit evaluation. Analytical automation techniques have increased the standardization of RBC index detection; however, indirect hematocrit measurements by blood analyzer, the automated HCT, do not correlate well with "micro-HCT" results in patients with hematological pathologies. We aimed to overcome those disadvantages in "micro-HCT" analysis using "ImageJ" processing software. Methods: 223 blood samples from the "general population" and 19 from sickle cell disease patients were examined in parallel for hematocrit values using the automated HCT, standard "micro-HCT," and "ImageJ" micro-HCT methods. Results: For the "general population" samples, the "ImageJ" values were significantly higher than the corresponding values evaluated by standard "micro-HCT" and automated HCT, except for the 0 to 2 month old newborns, in which the automated HCT results were similar to the "ImageJ" evaluated HCT. Similar to the "general population" cohort, we found significantly higher values measured by "ImageJ" compared to either "micro-HCT" or the automated HCT in SCD patients. Correspondent differences for the MCV and MCHC were also found. Discussion: This study introduces the "micro-HCT" assessment technique using the image-analysis module of "ImageJ" software. This procedure allows overcoming most of the data errors associated with the standard "micro-HCT" evaluation and can replace the use of complicated and expensive automated equipment. The presented results may also be used to develop new standards for calculating hematocrit and associated parameters for routine clinical practice. Keywords: Image analysis; Microcapillary hematocrit; RBC indices

On Coloring Resilient Graphs

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study $r$-resiliently $k$-colorable graphs, which are those $k$-colorable graphs that remain $k$-colorable even after the addition of any $r$ new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for $k$-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

Big Line Bundles over Arithmetic Varieties

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes