The distance function from a real algebraic variety

For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$.Comment: 24 pages, 4 figures, accepted for publication in Computer Aided Geometric Desig

Low Rank Vector Bundles on the Grassmannian G(1,4)

Here we define the concept of $L$-regularity for coherent sheaves on the Grassmannian G(1,4) as a generalization of Castelnuovo-Mumford regularity on ${\bf{P}^n}$. In this setting we prove analogs of some classical properties. We use our notion of $L$-regularity in order to prove a splitting criterion for rank 2 vector bundles with only a finite number of vanishing conditions. In the second part we give the classification of rank 2 and rank 3 vector bundles without "inner" cohomology (i.e. H^i_*(E)=H^i(E\otimes\Q)=0 for any $i=2,3,4$) on G(1,4) by studying the associated monads.Comment: 11 pages, no figure

Fel Oscillators with Tapered Undulators: Inclusion of Harmonic Generation and Pulse Propagation

We review the theory of FEL oscillators operating with tapered undulators. We consider the case of a uniform tapering and introduce a parameter which characterizes the effect of the tapering on the gain and on the saturation intensity. We analyze the effect of the tapering on the FEL dynamics by including the pulse propagation effects too. We analyze the importance of tapering as a tool to model the optical pulse shapes and to control the higher harmonic intensities

Quantum cryptography with an ideal local relay

We consider two remote parties connected to a relay by two quantum channels. To generate a secret key, they transmit coherent states to the relay, where the states are subject to a continuous-variable (CV) Bell detection. We study the ideal case where Alice's channel is lossless, i.e., the relay is locally situated in her lab and the Bell detection is performed with unit efficiency. This configuration allows us to explore the optimal performances achievable by CV measurement-device-independent (MDI) quantum key distribution (QKD). This corresponds to the limit of a trusted local relay, where the detection loss can be re-scaled. Our theoretical analysis is confirmed by an experimental simulation where 10^-4 secret bits per use can potentially be distributed at 170km assuming ideal reconciliation.Comment: in Proceedings of the SPIE Security + Defence 2015 conference on Quantum Information Science and Technology, Toulouse, France (21-24 September 2015) - Paper 9648-4

Asymptotics of degrees and ED degrees of Segre products

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface