Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in $\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s = o(1)$. Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm

The weakening relationship between the Impact Factor and papers' citations in the digital age

Historically, papers have been physically bound to the journal in which they were published but in the electronic age papers are available individually, no longer tied to their respective journals. Hence, papers now can be read and cited based on their own merits, independently of the journal's physical availability, reputation, or Impact Factor. We compare the strength of the relationship between journals' Impact Factors and the actual citations received by their respective papers from 1902 to 2009. Throughout most of the 20th century, papers' citation rates were increasingly linked to their respective journals' Impact Factors. However, since 1990, the advent of the digital age, the strength of the relation between Impact Factors and paper citations has been decreasing. This decrease began sooner in physics, a field that was quicker to make the transition into the electronic domain. Furthermore, since 1990, the proportion of highly cited papers coming from highly cited journals has been decreasing, and accordingly, the proportion of highly cited papers not coming from highly cited journals has also been increasing. Should this pattern continue, it might bring an end to the use of the Impact Factor as a way to evaluate the quality of journals, papers and researchers.Comment: 14 pages, 5 figure

Abelian Surfaces over totally real fields are Potentially Modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.Comment: 285 page