Counting Solutions of a Polynomial System Locally and Exactly

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a polydisc $\mathbf{\Delta}\subset\mathbb{C}^n$, our method aims to certify the existence of $k$ solutions (counted with multiplicity) within the polydisc. In case of success, it yields the correct result under guarantee. Otherwise, no information is given. However, we show that our algorithm always succeeds if $\mathbf{\Delta}$ is sufficiently small and well-isolating for a $k$-fold solution $\mathbf{z}$ of the system. Our analysis of the algorithm further yields a bound on the size of the polydisc for which our algorithm succeeds under guarantee. This bound depends on local parameters such as the size and multiplicity of $\mathbf{z}$ as well as the distances between $\mathbf{z}$ and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots in $\mathbf{\Delta}$ of the original system to the problem of solving a truncated system of degree $k$. In particular, if the multiplicity $k$ of $\mathbf{z}$ is small compared to the total degrees of the polynomials $f_i$, our method considerably improves upon known complete and certified methods. For the special case of a bivariate system, we report on an implementation of our algorithm, and show experimentally that our algorithm leads to a significant improvement, when integrated as inclusion predicate into an elimination method

Supersymmetry Breaking and alpha'-Corrections to Flux Induced Potentials

We obtain the vacuum solutions for M-theory compactified on eight-manifolds with non-vanishing four-form flux by analyzing the scalar potential appearing in the three-dimensional theory. Many of these vacua are not supersymmetric and yet have a vanishing three-dimensional cosmological constant. We show that in the context of Type IIB compactifications on Calabi-Yau threefolds with fluxes and external brane sources alpha'-corrections generate a correction to the supergravity potential proportional to the Euler number of the internal manifold which spoils the no-scale structure appearing in the classical potential. This indicates that alpha'-corrections may indeed lead to a stabilization of the radial modulus appearing in these compactifications.Comment: 28 pages, no figures, reference adde

Fiscal Competition in Space and Time: An Endogenous-Growth Approach

Is tax competition good for economic growth? The paper addresses this question by means of a simple model of endogenous growth. There are many small jurisdictions in a large federation and individual governments benevolently maximise the welfare of immobile residents. Investment is costly: Quadratic installation and de-installation costs limit the mobility of capital. The paper looks at optimal taxation and long-run growth. In particular, the effects of variations in the cost parameter on economic growth and taxation are considered. It is shown that balanced endogenous growth paths do not always exist and effects of changes in installation costs are ambiguous.

Fiscal Competition in Space and Time: An Endogenous-Growth Approach

Is tax competition good for economic growth? The paper addresses this question by means of a simple model of endogenous growth. There are many small jurisdictions in a large federation and individual governments benevolently maximise the welfare of immobile residents. Investment is costly: Quadratic installation and de-installation costs limit the mobility of capital. The paper looks at optimal taxation and long-run growth. In particular, the effects of variations in the cost parameter on economic growth and taxation are considered. It is shown that balanced endogenous growth paths do not always exist, that, if they exist, the economic growth rate is positively related to the mobility of capital, that the impact of the mobility prameter on the tax rate is ambiguous and that the tax rate may go to zero even if mobility costs are strictly positive.Fiscal Federalism, Tax Competition, Endogenous Growth

trackr: A Framework for Enhancing Discoverability and Reproducibility of Data Visualizations and Other Artifacts in R

Research is an incremental, iterative process, with new results relying and building upon previous ones. Scientists need to find, retrieve, understand, and verify results in order to confidently extend them, even when the results are their own. We present the trackr framework for organizing, automatically annotating, discovering, and retrieving results. We identify sources of automatically extractable metadata for computational results, and we define an extensible system for organizing, annotating, and searching for results based on these and other metadata. We present an open-source implementation of these concepts for plots, computational artifacts, and woven dynamic reports generated in the R statistical computing language

Real-space renormalization group flow in quantum impurity systems: local moment formation and the Kondo screening cloud

The existence of a length-scale $\xi_K\sim 1/T_K$ (with $T_K$ the Kondo temperature) has long been predicted in quantum impurity systems. At low temperatures $T\ll T_K$, the standard interpretation is that a spin-$\tfrac{1}{2}$ impurity is screened by a surrounding `Kondo cloud' of spatial extent $\xi_K$. We argue that renormalization group (RG) flow between any two fixed points (FPs) results in a characteristic length-scale, observed in real-space as a crossover between physical behaviour typical of each FP. In the simplest example of the Anderson impurity model, three FPs arise; and we show that `free orbital', `local moment' and `strong coupling' regions of space can be identified at zero temperature. These regions are separated by two crossover length-scales $\xi_{\text{LM}}$ and $\xi_K$, with the latter diverging as the Kondo effect is destroyed on increasing temperature through $T_K$. One implication is that moment formation occurs inside the `Kondo cloud', while the screening process itself occurs on flowing to the strong coupling FP at distances $\sim \xi_K$. Generic aspects of the real-space physics are exemplified by the two-channel Kondo model, where $\xi_K$ now separates `local moment' and `overscreening' clouds.Comment: 6 pages; 5 figure