Irreducible polynomials over F2r\mathbb{F}_{2^r} with three prescribed coefficients

For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz.Comment: Incorporated referee comments. Accepted for publication in Finite Fields App

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.Comment: Incorporated comments from referees. Accepted for publication in Int. J. Number Theor

Position angles and coplanarity of multiple systems from transit timing

Aims: We compare the apparent difference in timing of transiting planets (or eclipsing binaries) that are observed from widely separated locations (parallactic delay). Methods: A simple geometrical argument allow us to show that the apparent timing difference depends also on the on-sky position angle of the planetary (or secondary) orbit, relative to the ecliptic plane. Results: We calculate that on-sky position angle would be readily observable using the future PLATO and CHEOPS missions data, and possibility observable already in many known radial-velocity systems (if they exhibit transits). We also find that on-sky coplanarity of multiple objects in the same system can be probed more easily than the on-sky position angle of each of the objects separately. We calculate the magnitude of the effect for all currently known planets (should they exhibit transits), finding that almost 200 of them -- mostly radial-velocity detected planets -- have predicted timing effect larger than 1 second. We also compute the theoretical timing precision for the PLATO mission, that will observe a similar stellar population, and find that a 1 second effect would be frequently readily observable. We also find that on-sky coplanarity of multiple objects in the same system can be probed more easily than the on-sky position angle of each of the objects separately. Conclusions: We show a new observable from transit photometry becomes available when very high precision transit timing is available. We find that there is a good match between projected capabilities of the future space missions PLATO and CHEOPS and the new observable. We give some initial science question that such a new observable may be related to and help addressing.Comment: 4 pages, 2 figures. A&A accepte

Optimizing the search for transiting planets in long time series

Context: Transit surveys, both ground- and space- based, have already accumulated a large number of light curves that span several years. Aims: The search for transiting planets in these long time series is computationally intensive. We wish to optimize the search for both detection and computational efficiencies. Methods: We assume that the searched systems can be well described by Keplerian orbits. We then propagate the effects of different system parameters to the detection parameters. Results: We show that the frequency information content of the light curve is primarily determined by the duty cycle of the transit signal, and thus the optimal frequency sampling is found to be cubic and not linear. Further optimization is achieved by considering duty-cycle dependent binning of the phased light curve. By using the (standard) BLS one is either rather insensitive to long-period planets, or less sensitive to short-period planets and computationally slower by a significant factor of ~330 (for a 3yr long dataset). We also show how the physical system parameters, such as the host star's size and mass, directly affect transit detection. This understanding can then be used to optimize the search for every star individually. Conclusions: By considering Keplerian dynamics explicitly rather than implicitly one can optimally search the BLS parameter space. The presented Optimal BLS enhances the detectability of both very short and very long period planets while allowing such searches to be done with much reduced resources and time. The Matlab/Octave source code for Optimal BLS is made available.Comment: 7 pages, 4 figures, 1 table. A&A accepted. Source code is available at: http://www.astro.physik.uni-goettingen.de/~avivofir