The GHS attack is known to solve discrete logarithm problems (DLP) in the Jacobian of a curve C0 defined over the d degree extension field kd of k := Fq by mapping it to the DLP in the Jacobian of a covering curve C of C0 over k. Recently, classifications for all elliptic curves and hyperelliptic curves C0/kd of genus 2,3 which possess (2, ..., 2)-covering C/k of P were shown under an isogeny c...

A totally real polynomial in Z[x] with zeros α1 6 α2 6 · · · 6 αn has span αn − α1. Building on the classification of all characteristic polynomials of integer symmetric matrices having span less than 4, we obtain a classification of polynomials having span less than 4 that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. We obtain as characteristic poly...

1 Characteristic 0 We start with a basic question: given a polynomial f ∈ C[z1, · · · , zn] with a singularity at 0, how can we measure the “singularness” of this polynomial in a precise way? In other words: we can look at various singularities and see intuitively that some singularities are worse than others. For instance, it feels like a transverse self-intersection is probably not as bad as ...

Let F be a field of characteristic 0 or greater than d. Scharlau’s norm principle holds for separable field extensions K over F , for certain forms φ of degree d over F which permit composition. Introduction Let d ≥ 2 be an integer and let F be a field of characteristic 0 or > d. Let φ : V → F be a form of degree d on an F -vector space V of dimension n (i.e., after suitable identification, φ i...

One proves that a field carries compatible directed order if and only it has characteristic 0 is real or transcendence degree of at least 1 over the rational numbers.

in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

Let F be an algebraically closed field of characteristic 0 and f(x) a polynomial of degree strictly greater than one in F [x]. We show that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number of degree one polynomials with coefficients in F that commute with f . As a corollary, we obtain a theorem of E. A. Bertr...

We use the known linear lower bound for Tseitin's tautologies for establishing linear lower bounds on the degree of Nullstellensatz proofs (in the usual boolean setting) for explicitly constructed systems of polyno-mials of a constant (in our construction 6) degree. It holds over any eld of characteristic distinct from 2. Previously, a linear lower bound was proved 14] for an explicitly constru...

An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost’s cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras with symplectic involution.