Homogeneous Polynomials Of Degree Research Articles

On stability behaviors of 5D M-theory black objects

Using N = 2 supergravity formalism, we investigate certain behaviors of five-dimensional black objects from the compactification of M-theory on a Calabi–Yau three-fold. The manifold has been constructed as the intersection of two homogeneous polynomials of degrees (ω + 2, 1) and (2, 1) in a product of two weighted projective spaces given by WP4(ω,1,1,1,1)×P1 . First, we determine the allowed electric charge regions of the BPS and non BPS black holes obtained by wrapping M2-branes on appropriate two cycles in such a proposed Calabi–Yau three-fold. After that, we calculate the entropy of these solutions which takes a maximal value corresponding to ω = 1 defining the ordinary projective space P4 . For generic values of ω, we show that the non BPS states are unstable. Then, we conduct a similar study of five-dimensional black strings. Concerning the allowed magnetic charge regions of the BPS and non BPS black stringy solutions derived from M5-branes on dual divisors, we calculate the tension taking a minimal value for P4 . By determining the recombination factor, we show that the non-BPS black string states are stable in the allowed regions in the magnetic charge space.

-ZARISKI PAIRS OF SURFACE SINGULARITIES

Abstract Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$ . We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$ -invariant, but lie in distinct path-connected components of the $\mu ^*$ -constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$ , respectively) make a Zariski pair of curves in $\mathbb {P}^2$ , the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$ -Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

Projective joint spectra and characters of representations of [formula omitted

For a tuple of square complex-valued N×N matrices A1,…,An the determinant of their linear combination x1A1+⋯+xnAn, which is called a pencil, is a homogeneous polynomial of degree N in C[x1,...xn]. Zero-set of this polynomial is an algebraic set in the projective space CPn−1. This set is called the determinantal hypersurface or determinantal manifold of the tuple (A1,...,An). It was shown in [7] that determinantal hypersurfaces contain substantial information about representations of finite Coxeter groups. Namely, if G is a non-special Coxeter group of type A,B, or D, ρ1 and ρ2 are two linear representations of G, and the determinantal hypersurfaces of images of the Coxeter generators of G under ρ1 and ρ2 coincide as divisors in the projective space, the characters of ρ1 and ρ2 are equal, and, therefore, ρ1 and ρ2 are equivalent. In [30] this result was extended in the characters part to affine Coxeter groups of types B,C, and D. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of A type are not covered by this result, as their combinatorics is quite different. In this paper we explicitly construct a finite set in A˜n having the same property. We also show that every group which is a semidirect product of a finite group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensional representation of the group, the determinantal hypersurface of images of the set determines the representation character.

Geometry of elliptic normal curves of degree 6

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in . We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of . We study the images and of a sextic under the projection from a general point and a general line . In particular, we show that is k-normal for all and is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that is k-normal for all and is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4.

The isochronous center for Kukles homogeneous systems of degree eight

In this paper, we investigate Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015,2017). However, the proofs may have some gaps, hence they are still open. The gaps of Giné et al. (2015,2017) have been shown in Guo et al. (2021). In this paper, we consider isochronous centers in the family of Kukles homogeneous systems symmetric about the y-axis of degree eight. By using Period Abel constants and Gröbner basis of polynomial systems, we obtain that there is only one isochronous center excepting the linear center.

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations.

Stillman’s question for twisted commutative algebras

Let 𝔸n,m be the polynomial ring Sym(ℂn⊗ℂm) with the natural action of GLm(ℂ). We consider a family of GLm(ℂ)-stable ideals Jn,m in 𝔸n,m, each equivariantly generated by one homogeneous polynomial of degree 2 and show that the regularity of this family is unbounded. Using this, we negatively answer a question raised by Erman, Sam and Snowden on a generalization of Stillman’s conjecture.

Bounds on the torsion subgroup schemes of Néron–Severi group schemes

Let X↪Pr be a smooth projective variety defined by homogeneous polynomials of degree ≤d over an algebraically closed field. Let Pic X be the Picard scheme of X. Let Pic0X be the identity component of Pic X. The Néron–Severi group scheme of X is defined by NSX=(PicX)/(Pic0X)red. We give an explicit upper bound on the order of the finite group scheme (NSX)tor in terms of d and r. As a corollary, we give an upper bound on the order of the finite group πe´t1(X,x0)torab. We also show that the torsion subgroup (NSX)tor of the Néron–Severi group of X is generated by at most (deg⁡X−1)(deg⁡X−2) elements in various situations.

Equations for GL Invariant Families of Polynomials

We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from Question 7 in Ranestad and Sturmfels (Le Math. 75, 411–424, 2020) and Problem 13 in Sturmfels (2014), which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a Julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.

BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversion

We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of mathcal{N} = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of mathcal{N} = 2, D = 4 supergravity.

On centres and direct sum decompositions of higher degree forms

Higher degree forms are homogeneous polynomials of degree or equivalently symmetric d-linear spaces. This paper is mainly concerned about the algebraic structure of the centres of higher degree forms with applications specifically to direct sum decompositions, namely expressing higher degree forms as sums of forms in disjoint sets of variables. We show that the centre algebra of almost every form is the ground field, consequently almost all higher degree forms are absolutely indecomposable. If a higher degree form is decomposable, then we provide simple criteria and algorithms for direct sum decompositions by its centre algebra. It is shown that the direct sum decomposition problem can be boiled down to some standard tasks of linear algebra, in particular the computations of eigenvalues and eigenvectors. We also apply the structure results of centre algebras to provide a complete answer to the classical problem of whether a higher degree form can be reconstructed from its Jacobian ideal.

Varieties in Cages: A Little Zoo of Algebraic Geometry

A \(d^{\{n\}}\)-cage \(\mathsf K\) is the union of n groups of hyperplanes in \(\mathbb P^n\), each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing \(d^n\) points where hyperplanes from all groups intersect. These points are called the nodes of \(\mathsf K\). We study the combinatorics of nodes that impose independent conditions on the varieties \(X \subset \mathbb P^n\) containing them. We prove that if X, given by homogeneous polynomials of degrees \(\le d\), contains the points from such a special set \(\mathsf A\) of nodes, then it contains all the nodes of \(\mathsf K\). Such a variety X is very special: in particular, X is a complete intersection.

The normal reduction number of two-dimensional cone-like singularities

Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} = Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$.

Sharp affine isoperimetric inequalities for the volume decomposition functionals of polytopes

The nth power of the volume functional Vn of polytopes P in Rn, according to dimensions of the spaces spanned by any n unit outer normal vectors of P, is decomposed into n homogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

The structure of infinitesimal homeostasis in input\u2013output networks

Homeostasis refers to a phenomenon whereby the output x_o of a system is approximately constant on variation of an input {{mathcal {I}}}. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs {{mathcal {G}}} with a distinguished input node iota , a different distinguished output node o, and a number of regulatory nodes rho _1,ldots ,rho _n. In these models the input–output map x_o({{mathcal {I}}}) is defined by a stable equilibrium X_0 at {{mathcal {I}}}_0. Stability implies that there is a stable equilibrium X({{mathcal {I}}}) for each {{mathcal {I}}} near {{mathcal {I}}}_0 and infinitesimal homeostasis occurs at {{mathcal {I}}}_0 when (dx_o/d{{mathcal {I}}})({{mathcal {I}}}_0) = 0. We show that there is an (n+1)times (n+1)homeostasis matrixH({{mathcal {I}}}) for which dx_o/d{{mathcal {I}}}= 0 if and only if det (H) = 0. We note that the entries in H are linearized couplings and det (H) is a homogeneous polynomial of degree n+1 in these entries. We use combinatorial matrix theory to factor the polynomial det (H) and thereby determine a menu of different types of possible homeostasis associated with each digraph {{mathcal {G}}}. Specifically, we prove that each factor corresponds to a subnetwork of {{mathcal {G}}}. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of det (H) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of det (H). There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).

The isochronous centers for Kukles homogeneous system of degree nine

In this paper, we consider Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015) and Giné et al. (2017). However, the proofs may have some gaps, hence they are still open. In this paper, we consider isochronous centers in the family of Kukles homogeneous systems of degree nine. By using Period Abel constants and Gröbner basis of polynomial systems, we obtain that there is no isochronous center with the exception of the linear center.

Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

We classify the phase portraits in the Poincaré disc of the differential equations of the form \(x^{\prime } = -y + x f(x,y)\), \(\dot y =x + y f(x,y)\) where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2,3,4,5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2. All our results have been revised by the program P4; see Chaps. 9 and 10 of Dumortier et al. (UniversiText, Springer-Verlag, New York, 2006).

Horosphere-invariant mappings of the unit ball in [formula omitted

Horospheres, particular ellipsoids internally tangent to the boundary of the Euclidean unit ball B of Cn, play a pivotal role in the Denjoy–Wolff theory of holomorphic self-mappings of B. We use horospheres to introduce a new family of transforms on the class of normalized locally biholomorphic mappings of B into Cn in a vein similar to the Koebe transforms with respect to automorphisms of B. Like the Koebe transforms, these horosphere transforms preserve the property of convexity of a mapping. After studying the transforms with regard to convex mappings in particular, we give a broad characterization of all normalized holomorphic mappings that are invariant under the horosphere transforms. We also show that the notion of horosphere invariance for locally biholomorphic mappings is preserved by several commonly studied extension operators. We then apply our technique to provide a similar, simplified characterization of Koebe-invariant mappings. With these characterizations in hand, we show that, despite the existence of many mappings satisfying either type of invariance separately, the only normalized holomorphic mappings that enjoy both types of invariance are those mappings of the formf(z)=(z11−z1+Q(zˆ)(1−z1)2,zˆ(1−z1)) for z=(z1,zˆ)∈B, where zˆ=(z2,…,zn) and Q:Cn−1→C is a homogeneous polynomial of degree 2. For certain Q, these mappings are extreme points of the closed convex hull of the family of convex mappings of B.

On the uniqueness and expression of limit cycles in planar polynomial differential system via monotone iterative technique

ABSTRACT This paper investigates the limit cycles for planar differential system where are homogeneous polynomials of degree n. Using the method of upper and lower solutions we show that this system has a unique limit cycle under some suitable conditions. Moreover, by employing the monotone iterative technique, we provide a computational method to obtain the expression, or the asymptotical one, of this limit cycle in the polar form. The error estimation of the asymptotical solutions is also provided.

Towards the Complete Determination of Next-to-Minimal Weights of Projective Reed-Muller Codes

Projective Reed-Muller codes are obtained by evaluating homogeneous polynomials of degree d in $${\mathbb {F}}_q[X_0, \ldots , X_n]$$ on the points of a projective space of dimension n defined over a finite field $${\mathbb {F}}_q$$ . They were introduced by Lachaud, in 1986, and their minimum distance was determined by Serre and Sorensen. As for the higher Hamming weights, contributions were made by Rodier, Sboui, Ballet and Rolland, mostly for the case where $$d < q$$ . In 2016 we succeeded in determining all next-to-minimal weights when $$q = 2$$ , and in 2018 we determined all next-to-minimal weights for $$q = 3 $$ , and almost all of these weights for the case where $$q \ge 4$$ . In the present paper we determine some of the missing next-to-minimal weights of projective Reed-Muller codes when $$q \ge 4$$ . Our proofs combine results of geometric nature with techniques from Grobner basis theory.