32 research outputs found

### The symmetric signature

We define two related invariants for a $d$-dimensional local ring
$(R,\mathfrak{m},k)$ called syzygy and differential symmetric signature by
looking at the maximal free splitting of reflexive symmetric powers of two
modules: the top dimensional syzygy module $\mathrm{Syz}^d_R(k)$ of the residue
field and the module of K\"ahler differentials $\Omega_{R/k}$ of $R$ over $k$.
We compute these invariants for two-dimensional ADE singularities obtaining
$1/|G|$, where $|G|$ is the order of the acting group, and for cones over
elliptic curves obtaining $0$ for the differential symmetric signature. These
values coincide with the F-signature of such rings in positive characteristic.Comment: Shortened the proofs of Proposition 2.8 and Theorem 3.15; modified
Lemma 3.11; added Remark 3.6, Lemma 4.10, and Lemma 4.11; minor typos fixed;
improved exposition; updated reference

### F-signature function of quotient singularities

We study the shape of the F-signature function of a d-dimensional quotient singularity k\u301ax1,\u2026,xd\u301bG, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group G 86Gl(d,k). When G is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples

### A Pascal's theorem for rational normal curves

Pascal's Theorem gives a synthetic geometric condition for six points
$a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection
points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$,
$\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous
question in higher dimension: is there a coordinate-free condition for $d+4$
points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this
paper we find many of these conditions by writing in the Grassmann-Cayley
algebra the defining equations of the parameter space of $d+4$ ordered points
in $\mathbb{P}^d$ that lie on a rational normal curve. These equations were
introduced and studied in a previous joint work of the authors with
Giansiracusa and Moon. We conclude with an application in the case of seven
points on a twisted cubic.Comment: 16 pages, 1 figure. Comments are welcom

### Structure of CSS and CSS-T Quantum Codes

We investigate CSS and CSS-T quantum error-correcting codes from the point of
view of their existence, rarity, and performance. We give a lower bound on the
number of pairs of linear codes that give rise to a CSS code with good
correction capability, showing that such pairs are easy to produce with a
randomized construction. We then prove that CSS-T codes exhibit the opposite
behaviour, showing also that, under very natural assumptions, their rate and
relative distance cannot be simultaneously large. This partially answers an
open question on the feasible parameters of CSS-T codes. We conclude with a
simple construction of CSS-T codes from Hermitian curves. The paper also offers
a concise introduction to CSS and CSS-T codes from the point of view of
classical coding theory

### Solving degree, last fall degree, and related invariants

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Groebner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo-Mumford regularity

### Point configurations, phylogenetic trees, and dissimilarity vectors

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for
phylogenetic trees and asked two important questions about their relation to
the tropical Grassmannian. Multiple authors, using independent methods,
answered affirmatively the first of these questions, showing that dissimilarity
vectors lie on the tropical Grassmannian, but the second question, whether the
set of dissimilarity vectors forms a tropical subvariety, remained opened. We
resolve this question by showing that the tropical balancing condition fails.
However, by replacing the definition of the dissimilarity map with a weighted
variant, we show that weighted dissimilarity vectors form a tropical subvariety
of the tropical Grassmannian in exactly the way that Pachter--Speyer
envisioned. Moreover, we provide a geometric interpretation in terms of
configurations of points on rational normal curves and construct a finite
tropical basis that yields an explicit characterization of weighted
dissimilarity vectors.Comment: Final version. To appear in Proceedings of the National Academy of
Sciences of the United States of America (PNAS

### Thermal analysis of the antineutrino 144Ce source calorimeter for the SOX experiment

The technical note describes the calorimeter which will be used to measure the activity of the antineutrino 144Ce source of the SOX experiment at the Gran Sasso Laboratories. The principle of the calorimeter is based on the measurement of both mass flow and temperature increase of the water circulating in the heat exchanger surrounding the source. The calorimeter is vacuum insulated in order to minimize the heat losses. The preliminary design and thermal Finite Element Analysis (FEA) are reported in the note