Lacunaryx: Computing bounded-degree factors of lacunary polynomials

In this paper, we report on an implementation in the free software Mathemagix of lacunary factorization algorithms, distributed as a library called Lacunaryx. These algorithms take as input a polynomial in sparse representation, that is as a list of nonzero monomials, and an integer $d$, and compute its irreducible degree-$\le d$ factors. The complexity of these algorithms is polynomial in the sparse size of the input polynomial and $d$.Comment: 6 page

Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach

We present a new algorithm for the computation of the irreducible factors of degree at most $d$, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the computation of irreducible factors of degree at most $d$ of univariate lacunary polynomials and to the factorization of low-degree multivariate polynomials. The reduction runs in time polynomial in the size of the input polynomial and in $d$. As a result, we obtain a new polynomial-time algorithm for the computation of low-degree factors, with multiplicity, of multivariate lacunary polynomials over number fields, but our method also gives partial results for other fields, such as the fields of $p$-adic numbers or for absolute or approximate factorization for instance. The core of our reduction uses the Newton polygon of the input polynomial, and its validity is based on the Newton-Puiseux expansion of roots of bivariate polynomials. In particular, we bound the valuation of $f(X,\phi)$ where $f$ is a lacunary polynomial and $\phi$ a Puiseux series whose vanishing polynomial has low degree.Comment: 22 page

Bounded-degree factors of lacunary multivariate polynomials

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof

Evidence for thermal activation in the glassy dynamics of insulating granular aluminum conductance

Insulating granular aluminum is one of the proto-typical disordered insulators whose low temperature electrical conductance exhibits ubiquitous non-equilibrium phenomena. These include slow responses to temperature or gate voltage changes, characteristic field effect anomalies and ageing phenomena typical of a glass. In this system the influence of temperature on the glassy dynamics has remained elusive, leading to the belief that the slow relaxations essentially proceed via elastic quantum tunneling. A similar situation was met in insulating indium oxide and it was concluded that in high carrier density Anderson insulators, electrons form a quantum glass phase. In this work we experimentally demonstrate that thermal effects do play a role and that the slow dynamics in granular aluminum is subject to thermal activation. We show how its signatures can be revealed and activation energy distributions can be extracted, providing a promising grasp on the nature of the microscopic mechanism at work in glassy Anderson insulators. We explain why some of the experimental protocols previously used in the literature fail to reveal thermal activation in these systems. Our results and analyses call for a reassessment of the emblematic case of indium oxide, and question the existence of a quantum glass in any of the systems studied so far

Ageing and relaxation times in disordered insulators

We focus on the slow relaxations observed in the conductance of disordered insulators at low temperature (especially granular aluminum films). They manifest themselves as a temporal logarithmic decrease of the conductance after a quench from high temperatures and the concomitant appearance of a field effect anomaly centered on the gate voltage maintained. We are first interested in ageing effects, i.e. the age dependence of the dynamical properties of the system. We stress that the formation of a second field effect anomaly at a different gate voltage is not a "history free" logarithmic (lnt) process, but departs from lnt in a way which encodes the system's age. The apparent relaxation time distribution extracted from the observed relaxations is thus not "constant" but evolves with time. We discuss what defines the age of the system and what external perturbation out of equilibrium does or does not rejuvenate it. We further discuss the problem of relaxation times and comment on the commonly used "two dip" experimental protocol aimed at extracting "characteristic times" for the glassy systems (granular aluminum, doped indium oxide...). We show that it is inoperable for systems like granular Al and probably highly doped InOx where it provides a trivial value only determined by the experimental protocol. But in cases where different values are obtained like in lightly doped InOx or some ultra thin metal films, potentially interesting information can be obtained, possibly about the "short time" dynamics of the different systems. Present ideas about the effect of doping on the glassiness of disordered insulators may also have to be reconsidered.Comment: to appear in the proceedings of the 14th International Conference on Transport and Interactions in Disordered Systems (TIDS14

Electrical glassy behavior in granular aluminium thin films

We present new results obtained by field effect measurements on insulating granular Al thin films. First, reproducible and stable conductance fluctuations are seen in micron size samples as a function of gate voltage. The anomalous field effect and its slow relaxation already known to exist in macroscopic samples are shown to still exist in small samples and to have no influence on the fluctuations pattern. Secondly, "true" aging is demonstrated, i.e. the anomalous field effect relaxation depends on the time elapsed since the cooling, the longer this time the longer it takes for the system to react to a gate voltage change. Interpretations and implications of these findings are discussed.Comment: 5th International Conference on Electronic Crystals (ECRYS), Carg\`ese : France (2008

In brief... House Prices and School Quality: Evidence from State and Private Education in Paris

It is now widely understood that the quality of state schools in a neighbourhood has an impact on local house prices. Analysing data for Paris, Gabrielle Fack and Julien Grenet have looked deeper into this link by exploring how the presence of private schools influences parents' willingness to pay to live near good state schools.School catchment areas, France, private education, public education, housing, house prices

When do Better Schools Raise Housing Prices? Evidence from Paris Public and Private Schools

In this paper, we investigate how housing prices react to the quality of education offered by neighbouring public and private schools. The organization of secondary schooling in the city of Paris, which combines residence-based-assignment to public schools with a well-developed and almost entirely publicly funded private school system, offers a valuable empirical context for analyzing how private schools affect the capitalization of public school performance in housing prices. Using comprehensive data on both schools and real estate transact ions over the period 1997-2004, we develop a matching framework to carefully compare sales across school attendance boundaries. We find that a standard deviation increase in public school performance raises housing prices by 1.4 to 2.4%. Moreover, we show that the capitalization of public school performance in the price of real estate shrinks as the availability of private schools increases in the neighbourhood. Our results confirm the predictions of general equilibrium models of school choice that private schools, by providing an advantageous outside option to parents, tend to mitigate the impact of public school performance on housing prices.School attendance zones, private schools, housing markets,residential segregation

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B\"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solving, edited by Gurvits, Pebay, Rojas and Thompso

Determination of characteristic relaxation times and their significance in glassy disordered insulators

We revisit the field effect procedure used to characterise the slow dynamics of glassy Anderson insulators. It is shown that in the slowest systems the procedure fails and the "characteristic" time values extracted are not intrinsic but determined by the experimental procedure itself. In other cases (like lightly doped indium oxide) qualitative indications about the dynamics might be obtained, however the times extracted cannot be seen as characteristic relaxation times of the system in any simple manner, and more complete experiments are necessary. Implications regarding the effect of carrier concentration on the emergence of glassiness are briefly outlined.Comment: published with minor proof correction