Search

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by irreducible polynomials with square-free support have F-rational singularities, positively answering a question of Bath, Musta\c{t}\u{a}, and Walther.

We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction.

For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when $K$ has positive characteristic, the invariant ring is $F$-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for $\operatorname{SO}_t(K)$ as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example it readily yields the $a$-invariant and information on the Hilbert series. Indeed, we use this to show that the $h$-vector of the invariant ring for $\operatorname{SO}_t(K)$ need not be unimodal.

In 1965 Buchberger defined Gr\"obner bases and an algorithm to compute them. Despite a slow start, already in the eighties Gr\"obner bases had become the main device for symbolic computations involving polynomials as well as a theoretical tool for the investigation of ideals and varieties via the so-called Gr\"obner deformation techniques. Rings and algebraic varieties defined by means of determinants are among the most classical objects in commutative algebra, algebraic geometry and invariant theory. By the end of the the eighties the time was ripe for the computation of Gr\"obner bases of determinantal ideals. We will tell the tale of how (universal) Gr\"obner bases of determinantal ideals were identified and the key role played by Bernd Sturmfels and his collaborators in this enterprise., Comment: Written on the occasion of Bernd Sturmfels' sixtieth birthday conference (2022) and dedicated to him

Let $X$ be the union of $n$ generic linear subspaces of codimension $>1$ in $\mathbb{P}^d$. Improving an earlier bound due to Derksen and Sidman, we prove that the Castelnuovo-Mumford regularity of $X$ satisfies $ \operatorname{reg}(X) \le n - [n / (2d-1)]$., Comment: 21 pages

Let $R$ be a finitely generated $\mathbb N$-graded algebra domain over a Noetherian ring and let $I$ be a homogeneous ideal of $R$. Given $P\in Ass(R/I)$ one defines the $v$-invariant $v_P(I)$ of $I$ at $P$ as the least $c\in \mathbb N$ such that $P=I:f$ for some $f\in R_c$. A classical result of Brodmann asserts that $Ass(R/I^n)$ is constant for large $n$. So it makes sense to consider a prime ideal $P\in Ass(R/I^n)$ for all the large $n$ and investigate how $v_P(I^n)$ depends on $n$. We prove that $v_P(I^n)$ is eventually a linear function of $n$. When $R$ is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in a recent preprint., Comment: the final version of this note is going to appear in PAMS

We present a study of the damping and spin pumping properties of CoFeB/X/Pt systems with $\rm X=Al,Cr$ and $\rm Ta$. We show that the total damping of the CoFeB/Pt systems is strongly reduced when an interlayer is introduced independently of the material. Using a model that considers spin relaxation, we identify the origin of this contribution in the magnetically polarized Pt formed by the magnetic proximity effect (MPE), which is suppressed by the introduction of the interlayer. The induced ferromagnetic order in the Pt layer is confirmed by transverse magneto-optical Kerr spectroscopy at the M$_{2,3}$ and N$_7$ absorption edges as an element-sensitive probe. We discuss the impact of the MPE on parameter extraction in the spin transport model.

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Pad\'e approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fr\"oberg's Conjecture in commutative algebra., Comment: 20 pages

Nowadays Quantum Key Distribution represents the most mature quantum technology, and multiple countries as well as private institutions are building their quantum network. However, QKD devices are still far from representing a product within everyone's reach. Indeed, limitations in terms of compatibility with existing telecom infrastructure and limited performances in terms of secret key rate, using non-cryogenic detection systems, are still critical. In this work, we implemented a quantum key distribution link between Sicily (Italy) and Malta utilizing two different Single-Photon Avalanche Diode (SPAD) detectors. The performances of a standard commercial SPAD have been compared with the results achieved with a new prototype of fast-gated System in a Package (SiP) SPAD; the SiP detector has shown to be able to accomplish a fourteen times higher key rate compared with the commercial device over the channel showing 20 dB of losses.

The maximal minors of a matrix of indeterminates are a universal Gr\"obner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2 and Singular it extends the range of computability by at least one order of magnitude., Comment: Final version. To appear in the Journal of Symbolic Computation

Genomic surveillance of infectious diseases allows monitoring circulating and emerging variants and quantifying their epidemic potential. However, due to the high costs associated with genomic sequencing, only a limited number of samples can be analysed. Thus, it is critical to understand how sampling impacts the information generated. Here, we combine a compartmental model for the spread of COVID-19 (distinguishing several SARS-CoV-2 variants) with different sampling strategies to assess their impact on genomic surveillance. In particular, we compare adaptive sampling, i.e., dynamically reallocating resources between screening at points of entry and inside communities, and constant sampling, i.e., assigning fixed resources to the two locations. We show that adaptive sampling uncovers new variants up to five weeks earlier than constant sampling, significantly reducing detection delays and estimation errors. This advantage is most prominent at low sequencing rates. Although increasing the sequencing rate has a similar effect, the marginal benefits of doing so may not always justify the associated costs. Consequently, it is convenient for countries with comparatively few resources to operate at lower sequencing rates, thereby profiting the most from adaptive sampling. Finally, our methodology can be readily adapted to study undersampling in other dynamical systems.

We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of multidegrees in arbitrary positive multigraded settings. In these environments, we extend the notion of Cartwright-Sturmfels ideals by means of a standardization technique. Furthermore, we recover or extend important results in the literature, for instance: we provide a multidegree version of Hartshorne's result stating the upper semicontinuity of arithmetic degree under flat degenerations, and we give an alternative proof of Brion's result regarding multiplicity-free varieties., Comment: to appear in Advances in Mathematics

Can one tell if an ideal is radical just by looking at the degrees of the generators? In general, this is hopeless. However, there are special collections of degrees in multigraded polynomial rings, with the property that any multigraded ideal generated by elements of those degrees is radical. We call such a collection of degrees a radical support. In this paper, we give a combinatorial characterization of radical supports. Our characterization is in terms of properties of cycles in an associated labelled graph. We also show that the notion of radical support is closely related to that of Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded generators whose multidegrees form a radical support is a Cartwright-Sturmfels ideal. Conversely, a collection of degrees such that any multigraded ideal generated by elements of those degrees is Cartwright-Sturmfels is a radical support., Comment: 11 pages, 7 figures

We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation., Comment: To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

In this paper, we survey the theory of Cartwright-Sturmfels ideals. These are Z^n-graded ideals, whose multigraded generic initial ideal is radical. Cartwright-Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Groebner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright-Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright-Sturmfels.

This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of the ideals whose powers have a linear resolution in terms of the regularity of the Rees ring., Comment: The paper is dedicated to David Eisenbud on the occasion of his seventy-fifth birthday

We consider the optimal arrangement of two diffusion materials in a bounded open set $\Omega\subset \mathbb{R}^N$ in order to maximize the energy. The diffusion problem is modeled by the $p$-Laplacian operator. It is well known that this type of problems has no solution in general and then that it is necessary to work with a relaxed formulation. In the present paper we obtain such relaxed formulation using the homogenization theory, i.e. we replace both materials by microscopic mixtures of them. Then we get some uniqueness results and a system of optimality conditions. As a consequence we prove some regularity properties for the optimal solutions of the relaxed problem. Namely, we show that the flux is in the Sobolev space $H^1(\Omega)^N$ and that the optimal proportion of the materials is derivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has no solution in general. Our results extend those obtained by the first author for the Laplace operator.

Fluorescence microscopy and derived techniques are continuously looking for photodetectors able to guarantee increased sensitivity, high spatial and temporal resolution and ease of integration into modern microscopy architectures. Recent advances in single photon avalanche diodes (SPADs) fabricated with industry-standard microelectronic processes allow the development of new detection systems tailored to address the requirements of advanced imaging techniques (such as image-scanning microscopy). To this aim, we present the complete design and characterization of two bidimensional SPAD arrays composed of 25 fully independent and asynchronously-operated pixels, both having fill-factor of about 50% and specifically designed for being integrated into existing laser scanning microscopes. We used two different microelectronics technologies to fabricate our detectors: the first technology exhibiting very low noise (roughly 200 dark counts per second at room temperature), and the second one showing enhanced detection efficiency (more than 60% at a wavelength of 500 nm). Starting from the silicon-level device structures and moving towards the in pixel and readout electronics description, we present performance assessments and comparisons between the two detectors. Images of a biological sample acquired after their integration into our custom image-scanning microscope finally demonstrate their exquisite on-field performance in terms of spatial resolution and contrast enhancement. We envisage that this work can trigger the development of a new class of SPAD-based detector arrays able to substitute the typical single-element sensor used in fluorescence laser scanning microscopy., Comment: 15 pages, 7 figures, 3 tables

Traditional paradigms for imaging rely on the use of a spatial structure, either in the detector (pixels arrays) or in the illumination (patterned light). Removal of the spatial structure in the detector or illumination, i.e., imaging with just a single-point sensor, would require solving a very strongly ill-posed inverse retrieval problem that to date has not been solved. Here, we demonstrate a data-driven approach in which full 3D information is obtained with just a single-point, single-photon avalanche diode that records the arrival time of photons reflected from a scene that is illuminated with short pulses of light. Imaging with single-point time-of-flight (temporal) data opens new routes in terms of speed, size, and functionality. As an example, we show how the training based on an optical time-of-flight camera enables a compact radio-frequency impulse radio detection and ranging transceiver to provide 3D images., Comment: This is the final version as published in Optica Vol. 7, Issue 8, pp. 900-905 (2020)

The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases., Comment: Typos corrected. After the paper was published we came to know that our results have some overlap with the paper "Gr\"obner bases and the Cohen-Macaulay property of Li's double determinantal varieties" by Fieldsteel, Nathan; Klein, Patricia arXiv:1906.06817 published in Proc. Amer. Math. Soc. Ser. B 7 (2020), 142--158

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement., Comment: 15 pages, added a new section describing an irredundant primary decomposition of $J$

We present a study of the dynamic magnetic properties of TiN-buffered epitaxial thin films of the Heusler alloy Fe$_{1.5}$CoGe. Thickness series annealed at different temperatures are prepared and the magnetic damping is measured, a lowest value of $\alpha=2.18\times 10^{-3}$ is obtained. The perpendicular magnetic anisotropy properties in Fe$_{1.5}$CoGe/MgO are also characterized. The evolution of the interfacial perpendicular anisotropy constant $K^{\perp}_{\rm S}$ with the annealing temperature is shown and compared with the widely used CoFeB/MgO interface. A large volume contribution to the perpendicular anisotropy of $(4.3\pm0.5)\times 10^{5}$ $\rm J/m^3$ is also found, in contrast with vanishing bulk contribution in common Co- and Fe-based Heusler alloys., Comment: 5 pages, 5 figures

Real time, high resolution 3D reconstruction of scenes hidden from the direct field of view is a challenging field of research with applications in real-life situations related e.g. to surveillance, self-driving cars and rescue missions. Most current techniques recover the 3D structure of a non-lineof-sight (NLOS) static scene by detecting the return signal from the hidden object on a scattering observation area. Here, we demonstrate the full colour retrieval of the 3D shape of a hidden scene by coupling back-projection imaging algorithms with the high-resolution time-of-flight information provided by a single-pixel camera. By using a high effciency Single-Photon Avalanche Diode (SPAD) detector, this technique provides the advantage of imaging with no mechanical scanning parts, with acquisition times down to sub-seconds., Comment: 6 pages, 4 figures