Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Investigating the exceptionality of scattered polynomials

Abstract: Scattered polynomials over a finite field $\mathbb{F}_{q^n}$ have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-$q^t$-partially scattered and R-$q^t$-partially scattered polynomials, for $t$ a divisor of $n$. Indeed, a polynomial is scattered if and only if it is both L-$q^t$-partially scattered and R-$q^t$-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is is the hardest to be preserved is the L-$q^t$-partially scattered one. On the one hand, we are able to extend the classification results of exceptional scattered polynomials to exceptional L-$q^t$-partially scattered polynomials. On the other hand, the R-$q^t$-partially scattered property seems more stable. We present a large family of R-$q^t$-partially scattered polynomials, containing examples of exceptional R-$q^t$-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. In order to detect new examples of polynomials which are R-$q^t$-partially scattered, we introduce two different notions of equivalence preserving this property and concerning natural actions of the groups ${\rm \Gamma L}(2,q^n)$ and ${\rm \Gamma L}(2n/t,q^t)$. In particular, our family contains many examples of inequivalent polynomials, and geometric arguments are used to determine the equivalence classes under the action of ${\rm \Gamma L}(2n/t,q^t)$.

Deligne-Hodge-DeRham theory with coefficients

Abstract: Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $\Psi_f ({\cal L})$ on $Y$ can be described by these logarithmic techniques and a comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy which provides the link with various results on the subject.

Factoring and Testing Primes in Small Space

Abstract: We discuss how much space is sufficient to decide whether a unary number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. That is, un-Primes is in pebble-DSPACE(log log n) and also in accept-ASPACE(log log n), where un-primes={1 n :n is a prime}. Moreover, if the given n is composite, such machines are able to find a divisor of n. Since O(log log n) space is too small to write down a divisor which might require Ω(log n) bits, the witness divisor is indicated by the input head position at the moment when the machine halts.

Constants in Titchmarsh divisor problems for elliptic curves

Abstract: Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} \tau(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve $E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in the generic case of a Serre curve, we prove explicit closed formulae for $C(E)$ amenable to concrete computations. Moreover, we compute the moments of the constants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

Positivity of the height jump divisor

Abstract: We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight -1, pulled back along normal function sections. In the case of the normal function on M_g associated to the Ceresa cycle, our result proves a conjecture of Hain. As an application of our result we obtain that the Moriwaki divisor on \bar M_g has non-negative degree on all complete curves in \bar M_g not entirely contained in the locus of irreducible singular curves.