32 research outputs found

    The symmetric signature

    Full text link
    We define two related invariants for a dd-dimensional local ring (R,m,k)(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 SyzRd(k)\mathrm{Syz}^d_R(k) of the residue field and the module of K\"ahler differentials ΩR/k\Omega_{R/k} of RR over kk. We compute these invariants for two-dimensional ADE singularities obtaining 1/∣G∣1/|G|, where ∣G∣|G| is the order of the acting group, and for cones over elliptic curves obtaining 00 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

    Get PDF
    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

    Get PDF
    Pascal's Theorem gives a synthetic geometric condition for six points a,…,fa,\ldots,f in P2\mathbb{P}^2 to lie on a conic. Namely, that the intersection points ab‾∩de‾\overline{ab}\cap\overline{de}, af‾∩dc‾\overline{af}\cap\overline{dc}, ef‾∩bc‾\overline{ef}\cap\overline{bc} are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d+4d+4 points in Pd\mathbb{P}^d to lie on a degree dd 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+4d+4 ordered points in Pd\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

    Full text link
    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

    Get PDF
    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

    Full text link
    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

    Get PDF
    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
    corecore