Odd Degree Research Articles

On hyperelliptic curves of odd degree and genus g with 6 torsion points of order 2g + 1

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0, given by the equation y2=f(x), where the polynomial f(x)∈K[x]is square-free and has odd degree 2g+1. The curve Ccontains a single “infinite” point O, which is the Weierstrass point. There is a classical embedding of C(K)into the group of K-points J(K)of the Jacobian variety Jof the curve C, identifying the point Owith the unit element of the group J(K). For 2≤g≤5, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves Cwith a marked unique point at infinity Othat the set C(K)∩J(K)contains at least 6 torsion points of order 2g+1. It was previously known that for g=2there are exactly 5 such equivalence classes, and for g≥3an upper bound was known that depended only on the genus of g. We improve the previously known upper bound by almost 36 times.

Kernels of minimal characters of solvable groups

Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

A parametrization of nonassociative cyclic algebras of prime degree

We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree m when the base field contains a primitive mth root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field F, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.

An optimal ansatz space for moving least squares approximation on spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere Sd-1⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb S^{d-1} \\subset {\\mathbb R}^d$$\\end{document}, where d>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>1$$\\end{document}. It is well known that using the spherical harmonics up to degree L∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L \\in {\\mathbb N}$$\\end{document} as ansatz space yields for functions in CL+1(Sd-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}^{L+1}(\\mathbb S^{d-1})$$\\end{document} the approximation order OhL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( h^{L+1} \\right) $$\\end{document}, where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\rightarrow 0$$\\end{document}. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.

On Hyperelliptic Curves of Odd Degree and Genus g with Six Torsion Points of Order 2g + 1

On Hyperelliptic Curves of Odd Degree and Genus g with Six Torsion Points of Order 2g + 1

Positivity and Entanglement of Polynomial Gaussian Integral Operators

Abstract Positivity preservation is an important issue in the dynamics of open quantum systems: positivity violations always mark the border of validity of the model. We investigate the positivity of self-adjoint polynomial Gaussian integral operators $\widehat{\kappa }_{\operatorname{PG}}$; i.e. the multivariable kernel $\kappa _{\operatorname{PG}}$ is a product of a polynomial $P$ and a Gaussian kernel $\kappa _G$. These operators frequently appear in open quantum systems. We show that $\widehat{\kappa }_{\operatorname{PG}}$ can only be positive if the Gaussian part is positive, which yields a strong and quite easy test for positivity. This has an important corollary for the bipartite entanglement of the density operators $\widehat{\kappa }_{\operatorname{PG}}$: if the Gaussian density operator $\widehat{\kappa }_G$ fails the Peres–Horodecki criterion, then the corresponding polynomial Gaussian density operators $\widehat{\kappa }_{\operatorname{PG}}$ also fail the criterion for all $P$; hence they are all entangled. We prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite. We introduce a new preorder $\preceq$ on Gaussian kernels such that if $\kappa _{G_0}\preceq \kappa _{G_1}$ then $\widehat{\kappa }_{\operatorname{PG}_0}\ge 0$, implying that $\widehat{\kappa }_{\operatorname{PG}_1}\ge 0$ for all polynomials $P$. Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of other polynomial Gaussian operators having the same polynomial factor, which might improve any given positivity test by carrying it out on a much larger set of operators. We will show an example that this really can make positivity tests much more sensitive and efficient. This preorder has implications for the entanglement problem, too.

On the construction of certain odd degree irreducible polynomials over finite fields

On the construction of certain odd degree irreducible polynomials over finite fields

Odd and Co – Odd Sum Degree Domination in Graphs with Algorithm

Odd and Co – Odd Sum Degree Domination in Graphs with Algorithm

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

Solubility of additive forms of twice odd degree over totally ramified extensions of Q2$\mathbb {Q}_2$

AbstractWe prove that an additive form of degree , odd over any totally ramified extension of has a nontrivial zero if the number of variables satisfies .

Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

Odd Wilson surfaces

Previously, Wilson surface observables were interpreted as a class of Poisson sigma models. We profit from this construction to define and study the super version of Wilson surfaces. We provide some ‘proof of concept’ examples to illustrate modifications resulting from appearance of odd degrees of freedom in the target.

Odd degree rational characters and the order of rational elements in finite groups

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by P. H. Tiep and H. P. Tong-Viet.

Maximal Antipodal Subgroups and Covering Homomorphisms with Odd Degree

We show that all of maximal antipodal subgroups in compact Lie groups, which are not necessarily connected, do not change through covering homomorphisms with odd degree.

A Viro–Zvonilov-type inequality for Q-flexible curves of odd degree

A Viro–Zvonilov-type inequality for Q-flexible curves of odd degree

Reaction-diffusion equations on metric graphs with edge noise

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.

Most odd-degree binary forms fail to primitively represent a square

Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

The Coble quadric

Abstract Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in ${\mathbb {P}}^8$ as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$ , the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2,8)$ . In fact, each point $p\in C$ defines a natural embedding of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$ . We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$ .

Hierarchical stability conditions for linear systems with interval time-varying delay

This paper proposes the hierarchical stability conditions for linear systems with an interval time-varying delay. In the matter of the delay, the upper and lower bounds are restricted, but there is no constraint on its derivative. Firstly, based on the state vectors and multiple integral state vectors involved in the delay-related generalized free-matrix-based integral inequalities (GFIIs), the hierarchical Lyapunov-Krasovskii functional (LKF) candidates are constructed. Then, the GFIIs are applied for the approximation of the integral quadratic terms, which will introduce the delay related nonlinear terms in the LKF differential. Next, the novel adjusted matrix-valued high and odd degree polynomial negative definite conditions (NDCs) are provided to acquire the hierarchical linear matrix inequality (LMI) conditions and cope with the nonlinear terms introduced by the GFIIs. Eventually, the advantages of the obtained stability conditions are checked through some numeric examples.

Odd degree isolated points on $$X_1(N)$$ with rational j-invariant

Odd degree isolated points on $$X_1(N)$$ with rational j-invariant