On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset{\mathbb R}^m$ and ${\mathcal S}^*(M)$ its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps ${\rm Spec}({\tt j})_1:{\rm Spec}({\mathcal S}(N))\to{\rm Spec}({\mathcal S}(M))$ and ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$ induced by the inclusion ${\tt j}:N\hookrightarrow M$ of a semialgebraic subset $N$ of $M$. The ring ${\mathcal S}(M)$ can be understood as the localization of ${\mathcal S}^*(M)$ at the multiplicative subset ${\mathcal W}_M$ of those bounded semialgebraic functions on $M$ with empty zero set. This provides a natural inclusion ${\mathfrak i}_M:{\rm Spec}({\mathcal S}(M))\hookrightarrow{\rm Spec}({\mathcal S}^*(M))$ that reduces both problems above to an analysis of the fibers of the spectral map ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$. If we denote $Z:={\rm cl}_{{\rm Spec}({\mathcal S}^*(M))}(M\setminus N)$, it holds that the restriction map ${\rm Spec}({\tt j})_2|:{\rm Spec}({\mathcal S}^*(N))\setminus{\rm Spec}({\tt j})_2^{-1}(Z)\to{\rm Spec}({\mathcal S}^*(M))\setminus Z$ is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of ${\rm Spec}({\tt j})_2$ at the points of $Z$. The size of the fibers of prime ideals `close' to the complement $Y:=M\setminus N$ provides valuable information concerning how $N$ is immersed inside $M$. If $N$ is dense in $M$, the map ${\rm Spec}({\tt j})_2$ is surjective and the generic fiber of a prime ideal ${\mathfrak p}\in Z$ contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber ${\rm Spec}({\tt j})_2^{-1}({\mathfrak p})$ is a finite set for ${\mathfrak p}\in Z$.Comment: 33 pages, 3 figure

On the one dimensional polynomial and regular images of Rn\R^n

In this work we present a full geometric characterization of the 1-dimensional polynomial and regular images $S$ of $\R^n$ and we compute for all of them the invariants ${\rm p(S)}$ and ${\rm r}(S)$, already introduced in \cite{fg2}

Tuning High-Harmonic Generation by Controlled Deposition of Ultrathin Ionic Layers on Metal Surfaces

High harmonic generation (HHG) from semiconductors and insulators has become a very active area of research due to its great potential for developing compact HHG devices. Here we show that by growing monolayers (ML) of insulators on single-crystal metal surfaces, one can tune the harmonic spectrum by just varying the thickness of the ultrathin layer, not the laser properties. This is shown from numerical solutions of the time-dependent Schr\"odinger equation for $n$ML NaCl/Cu(111) systems ($n=1-50$) based on realistic potentials available in the literature. Remarkably, the harmonic cutoff increases linearly with $n$ and as much as an order of magnitude when going from $n$ $=$ 1 to 30, while keeping the laser intensity low and the wavelength in the near-infrared range. Furthermore, the degree of control that can be achieved in this way is much higher than by varying the laser intensity. The origin of this behavior is the reduction of electronic "friction" when moving from the essentially discrete energy spectrum associated with a few-ML system to the continuous energy spectrum (bands) inherent to an extended periodic system.Comment: 6 pages, 4 figure