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Assume that we are given a sequence F of k homogeneous polynomials in n variables of degree at most d and the ideal ℐ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity d O (n) , whether F is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if F is regular then we can transform ℐ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity d O (n 2) . Finally, under the same assumption, we establish the new upper bound 2 (d k / 2) 2 n − k − 1 for the maximum degree of the elements of any reduced Gröbner basis of ℐ in the case that n > k. [ABSTRACT FROM AUTHOR]

In this paper, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by Fonseca and Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution. [ABSTRACT FROM AUTHOR]

In this paper, an observer-based tracking control problem is considered for a polynomial system with parameter uncertainties and external disturbances. Noting that some states are difficult to measure in practice, a polynomial state observer is employed to obtain the unknown states. Then a robust observer-based tracking controller is designed to force the states of the polynomial uncertain system to follow the ones of the reference model and satisfy $ H_\infty $ H ∞ performance. By using a full-state-dependent homogeneous polynomial function, the solvability condition is derived for the observer-based tracking control strategy, which reduces conservatism. Sum of squares technique is utilised to calculate the corresponding observer and controller, effectively dealing with the calculation problem in polynomial systems. Simulation examples are given to illustrate the effectiveness of the presented control strategy. [ABSTRACT FROM AUTHOR]

A serious threat to filter design is the difficulty in resisting important‐data‐based (IDB) Denial of Service (DoS) attacks. In this article, a new approach to filter design that is more relaxed and flexible in resisting IDB DoS policies is proposed. The proposed approach introduces the technique of homogeneous polynomials to develop a more general relaxed fuzzy filter design that can reduce conservatism. Moreover, it is guaranteed that the new method will obtain sets of more relaxed filter gains, which are more capable of responding to the IDB DoS policy. This enables the filter to perform more accurate signal estimation and ensures the filtering error system's H∞$$ {H}_{\infty } $$ performance. Finally, the reliability and effectiveness of the proposed filter method are demonstrated by two comparative examples. [ABSTRACT FROM AUTHOR]

We prove that any planar polynomial differential system with linear plus homogeneous nonlinearities has no isochronous global centers and obtain all cubic polynomial differential systems having an isochronous global center. [ABSTRACT FROM AUTHOR]

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with, but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)‐module. The main aim of the paper is to explain that a good possibility is to take the so‐called generalized Cauchy–Riemann equations proposed a long time ago by Stein and Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)‐modules. This is an important step in developing the corresponding function theory. [ABSTRACT FROM AUTHOR]

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into \mathbb P^2 intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f. As applications, we study some cases of Campana's orbifold conjecture for \mathbb P^2 and finite ramified covers of \mathbb P^2 with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of \mathbb G_m^2. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

Homogeneous polynomial functions have the potential to provide a general modeling framework for yield surfaces in metal plasticity. They incorporate as particular cases many of the previously proposed yield functions and their fitting capabilities allow for capturing a wide range of yield surface shapes. And yet, there are still two unsolved problems which turn into major obstacles when it comes to actual implementations in both academic and industrial environments: The lack of a general optimization algorithm for the calibration of their parameters and the lack of an efficient computational scheme for their value, gradient and hessian. The difficulty of the first problem is two-fold, necessitating an adequate specification of the experimental input data set and satisfaction of the convexity constraint. The second problem is specific to all high degree polynomials and is comprised of issues such as numerical stability, precision and implementation efficiency. We present practical solutions to both problems: An optimization algorithm that reduces to solving a sequence of quadratic problems and a double Horner evaluation scheme that is optimal (featuring the least number of multiplications). The resulting modeling framework can account for arbitrary input data, experimental or from crystal plasticity predictions. As illustration we show new results regarding the relationship between generalized r-values and the earing profile of deep-drawn cylindrical cups. Practicality is demonstrated by the high level of automation of the entire workflow, from material parameters calibration to finite element simulations, and supporting code (Python scripts and constitutive subroutine) made available at https://github.com/stefanSCS/PolyN. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork's theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$ , which computes the rigid cohomology of $\mathbb {P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz's comparison results in [19] for projective hypersurface complements to arbitrary projective complements. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to exhibit failure prediction capability of polynomial-based yield functions with a basic damage model. For this purpose, a constitutive model considering anisotropic plasticity and ductile fracture was developed. In this model, anisotropic plastic behavior of dual phase steels, namely DP600 and DP800, was described by quadratic Hill48 and non-quadratic anisotropic homogeneous the fourth-order polynomial (HomPol4) stress potentials and the generalized plastic work criterion from ductile damage models was used for the prediction of fracture initiation. The model has been implemented into an implicit finite element (FE) code. The parameters of the constitutive model were calibrated with uniaxial tensile tests performed in different directions with respect to the rolling direction of the materials and anisotropic stress potentials were evaluated by comparison of the predicted in-plane variations of the plastic properties (yield stress ratios and Lankford coefficients), and yield locus contours with experimental data. The calibrated model was firstly applied to uniaxial tensile test and then to a hole expansion test to predict fracture. The stroke values at fracture, hole expansion ratios (HER) and fracture locations were investigated. Any significant difference between the anisotropic stress potentials was not observed in terms of HER predictions, however plastic work criterion in conjunction with HomPol4 function predicted the crack initiation locations accurately on the fractured samples. Afterward, the Lode parameter and stress triaxiality effects were investigated in fracture stroke prediction. Since the HomPol4 predictions of fracture initiation locations are accurate, the predicted HomPol4 results from the generalized plastic work criterion were compared with the modified Mohr-Coulomb ductile fracture model results. A significant improvement was observed in the fracture displacement predictions. However, it is seen that the failure location predictions of both models were the same. From these results, it can be concluded that the HomPol4 yield criterion has an effective potential to predict the failure locations even though with a basic damage model. In the current study, the out-of-plane anisotropy effect was assessed as well. To this end, Hill48's parameter correlated with the out-of-plane shear components were adjusted. It was found that the out-of-plane anisotropy has a negligible effect on the predictions of HER and fracture initiation location. [ABSTRACT FROM AUTHOR]

We deal with Perazzo hypersurfaces X = V (f) in P n + 2 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v) = p 0 (u , v) x 0 + p 1 (u , v) x 1 + ⋯ + p n (u , v) x n + g (u , v) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v ] and g is a form in K [ u , v ] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra A f fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of A f , we prove that the Hilbert function of A f is always unimodal and we determine when A f satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V (f) in P n + 3 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v , w) = p 0 (u , v , w) x 0 + p 1 (u , v , w) x 1 + ⋯ + p n (u , v , w) x n + g (u , v , w) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v , w ] and g is a form in K [ u , v , w ] of degree d. [ABSTRACT FROM AUTHOR]

We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gröbner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity. [ABSTRACT FROM AUTHOR]

This note addresses the problem of recovering a convex set of homogeneous polynomials from the subset of its extreme points, i.e., the justification of a polynomial version of the classical Krein–Milman theorem. Not much was done in this direction. The existing papers deal mostly with the description of the extreme points of the unit ball in the space of homogeneous polynomials in various special cases. Even in the case of linear operators, the classical Krein–Milman theorem does not work, since closed convex sets of operators turn out to be compact in some natural topology only in rather special cases. In the 1980s, a new approach to the study of the extremal structure of convex sets of linear operators was proposed on the basis of the theory of Kantorovich spaces, which led to an operator form of the Krein–Milman theorem. Combining the approach with the linearization method for homogeneous polynomials, we obtain a version of the Krein–Milman theorem for homogeneous polynomials. Namely, a weakly order bounded, operator convex, and pointwise order closed set of homogeneous polynomials from an arbitrary vector space to a Kantorovich space is the pointwise order closure of the operator convex hull of the extreme points of . We also establish Milman's converse of the Krein–Milman theorem for homogeneous polynomials: The extreme points of the smallest operator convex pointwise order closed set including a given set of homogeneous polynomials are pointwise uniform limits of appropriate nets of mixings in . A mixing of a family of polynomials with the values in a Kantorovich space is understood as the (infinite) sum of these polynomials multiplied by pairwise disjoint order projections with sum the identity operator in the Kantorovich space. [ABSTRACT FROM AUTHOR]

Let G = Cp × H be a finite group, where Cp is a cyclic group of prime order p and H is a p'-group. Let F be an algebraically closed field in characteristic p. Let V be a direct sum of m non-trivial indecomposable G-modules such that the norm polynomials of the simple H-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring F[V] with actions of G. In this paper, by the periodicity property, we showed that F[V]G is generated by m norm polynomials together with homogeneous invariants of degree at most m|G| - dim(V) and transfer invariants, which yields the well-known degree bound dim(V)·(|G|-1). More precisely, we found that this bound gets less sharp as the dimensions of simple H-modules increase. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

The Friedrichs operator of a domain (in \mathbb {C}^n) is closely related to its Bergman projection and encodes crucial information (geometric, quadrature, potential theoretic etc.) about the domain. We show that the Friedrichs operator of a domain has rank one if the domain can be covered by a circular domain via a proper holomorphic map of finite multiplicity whose Jacobian is a homogeneous polynomial. As an application, we show that the Friedrichs operator is of rank one on the tetrablock, pentablock, and the symmetrized polydisc – domains of significance in the study of \mu-synthesis in control theory. [ABSTRACT FROM AUTHOR]

We give an intrinsic projective configuration of the maximal hypersurfaces of degree d in P n (F q) . It is a particular arrangement of hyperplanes containing subarrangements of hyperplanes in general position. This characterization allows us to compute the number of minimum distance codewords of the projective Reed-Muller code PRM(q, d, n). [ABSTRACT FROM AUTHOR]

We study properties of the eigenvalues of the Laplace–Beltrami operator on L2 S d N - 1 , restricted to the class of antisymmetric functions. We prove that Vandermonde type determinants generate a basis for the space of antisymmetric homogeneous harmonic polynomials. Based on this fact, we calculate the multiplicity of each eigenvalue for d = 1 and present an algorithm for calculating eigenvalues for d ⩾ 2. [ABSTRACT FROM AUTHOR]

For the nonlinear parameter‐varying (NPV) model of unmanned surface vehicle with the consideration of the velocities on yaw and surge as well as wave disturbances, a robust H∞$$ {H}_{\infty } $$ control method is proposed based on extended homogeneous polynomial Lyapunov function (EHPLF) to regulate heading for the superior performance on the rapidity, accuracy, and robustness. First, a NPV model of heading error is established to design a general form of a state feedback controller with a robust H∞$$ {H}_{\infty } $$ performance. Second, a Lyapunov matrix with full states and varying parameter is constructed to derive the robust H∞$$ {H}_{\infty } $$ global exponential stability conditions by Euler's homogeneity relation for the NPV system, known as the EHPLF stability conditions. Third, since the EHPLF stability conditions consist of a set of nonlinear coupled inequalities that cannot be directly solved by sum of squares (SOS) toolboxes, they are decoupled with matrix transformations to obtain the EHPLF‐SOS stability conditions, which is solved for the parameters of the state feedback controller. Finally, the simulation results indicate that EHPLF method exhibits a superior performance on dynamic, steady‐state, and robustness. [ABSTRACT FROM AUTHOR]

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, Z 2 ⊕ Z 2 symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits. [ABSTRACT FROM AUTHOR]

We begin with a degree m real homogeneous polynomial in n indeterminants and bound the distance from a given n-dimensional real vector to the real vanishing of the homogeneous polynomial. We then apply these bounds to the real homogeneous polynomial associated with a nonzero m-order n-dimensional weakly symmetric tensor which has zero as an eigenvalue. We provide "nested spheres" conditions to bound the distance from a given n-dimensional real vector to the nearest zero eigenvector. [ABSTRACT FROM AUTHOR]

Herein, a pure shifted Jacobi–Galerkin (SJG) method is offered for handling linear two-dimensional space-time diffusion and telegraph equations but by considering their integrated forms. The semi-analytic approximate solution is expanded in spatial and temporal variables as polynomials bases built as a linear combination of the shifted Jacobi Polynomials (JPs). Expanding the exact solution regarding these polynomials satisfies the homogeneous initial and boundary conditions. The proposed Jacobi–Galerkin (JG) procedure yields exponential convergence rates when the solution is smooth enough. A careful investigation of the convergence analysis of the suggested approximate triple series expansion examines the offered numerical algorithm. Two test illustrative examples are given to prove the high accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths. [ABSTRACT FROM AUTHOR]

We construct a system of solutions for the coefficients of expansion in the system of trigonometric functions for the solution of the Helmholtz equation in a cylindrical coordinate system in the form of homogeneous polynomials in two biorthogonal systems of functions. Some properties of the biorthogonal systems of functions are proved. [ABSTRACT FROM AUTHOR]

We study Hermite interpolation for the space of polynomials of total degree in ℝN and the space of homogeneous polynomials in ℝN+1. We investigate the relations between the two types of Hermite interpolation. We show that they have the same regularity and continuity property. [ABSTRACT FROM AUTHOR]

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also establish a sharper result for this kind for homogeneous polynomials, assuming that the characteristic does not divide the degree. Further, we use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan–Hochster's theorem [J. Amer. Math. Soc., 33, No. 1, 291–309 (2020), Theorem A]. [ABSTRACT FROM AUTHOR]

We introduce the universal algebra of two Poisson algebras P and Q as a commutative algebra A : = P (P , Q) satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P -module U , we construct a functor U ⊗ − : A M → Q P M from the category of A -modules to the category of Poisson Q -modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A -module, then there exists another functor − ⊗ V : P P M → Q P M connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n -dimensional, then P (P) : = P (P , P) is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, P (P) can be described as the quotient of the polynomial algebra k [ X i j | i , j = 1 , ⋯ , n ] through an ideal generated by 2 n 3 non-homogeneous polynomials of degree ≤2. Two applications are provided. The first one describes the automorphisms group Aut Poiss (P) as the group of all invertible group-like elements of the finite dual P (P) o. Secondly, we show that for an abelian group G , all G -gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra P (P). [ABSTRACT FROM AUTHOR]

To any n-curve U(r) (i.e. a plane curve represented by a homogeneous polynomial of degree n in r = (r1, r2, r3)) and arbitrary point X one can associate a set of n lines (a.k.a. a pencil) through X. Varying X produces a family of pencils and these define an envelope (the dual of U) and this dual curve is generally a curve of degree n(n-1). This result can be extended to consider two conics (resulting in the harmonic envelope). It is shown here that other combinations of curves (specifically using two cubics or three conics) may also be used to construct curves with interesting properties. [ABSTRACT FROM AUTHOR]

We prove that over any perfect field, the plane Cremona group is generated by involutions. [ABSTRACT FROM AUTHOR]

A function f$$ f $$ from a domain in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$ to the quaternions is said to be inframonogenic if ∂‾f∂‾=0$$ \overline{\partial}f\overline{\partial}=0 $$, where ∂‾=∂/∂x0+(∂/∂x1)e1+(∂/∂x2)e2$$ \overline{\partial}=\partial /\partial {x}_0+\left(\partial /\partial {x}_1\right){e}_1+\left(\partial /\partial {x}_2\right){e}_2 $$. All inframonogenic functions are biharmonic. In the context of functions f=f0+f1e1+f2e2$$ f={f}_0+{f}_1{e}_1+{f}_2{e}_2 $$ taking values in the reduced quaternions, we show that the inframonogenic homogeneous polynomials of degree n$$ n $$ form a subspace of dimension 6n+3$$ 6n+3 $$. We use the homogeneous polynomials to construct an explicit, computable orthogonal basis for the Hilbert space of square‐integrable inframonogenic functions defined in the ball in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$. [ABSTRACT FROM AUTHOR]

A centre of a differential system in the plane R 2 is a singular point p having a neighbourhood U such that U ∖ { p } is filled of periodic orbits. A global centre is a centre p such that R 2 ∖ { p } is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (1) x ˙ = − y + P (x , y) , a ˙ y = x + Q (x , y) , where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups. [ABSTRACT FROM AUTHOR]

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields. [ABSTRACT FROM AUTHOR]

Let F_1, ..., F_R be homogeneous polynomials of degree d\geqslant 2 with integer coefficients in n variables, and let \mathbf {F}=(F_1,\ldots,F_R). Suppose that F_1, ..., F_R is a non-singular system and n\geqslant 4^{d+2}d^2R^5. We prove that there are infinitely many solutions to \mathbf {F}(\mathbf {x})=\mathbf {0} in prime coordinates if (i) \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution over the p-adic units \mathbb {U}_p for all prime numbers p, and (ii) \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution in the open cube (0,1)^n. [ABSTRACT FROM AUTHOR]

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in P 5 . We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of I ( Sec (C 6)) . We study the images C p and C p q of a sextic C 6 under the projection from a general point P ∈ P 5 and a general line P Q ¯ ⊂ P 5 . In particular, we show that C p is k-normal for all k ≥ 2 and I (C p) is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that C p q is k-normal for all k ≥ 3 and I (C p q) is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4. [ABSTRACT FROM AUTHOR]

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension's irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions. [ABSTRACT FROM AUTHOR]

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet potential to be bounded from below by using the co-positivity of tensors. This is achieved by treating the potential as a fourth-order homogeneous polynomial about the moduli of the two Higgs doublet fields, where the "angles" is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment. Finally, the analytic conditions are established with the help of the corresponding theory and methods of higher-order tensors. [ABSTRACT FROM AUTHOR]

Let L be a second-order elliptic operator with analytic coefficients defined in B1 ⊆ ℝ. We construct explicitly and canonically a fundamental solution for the operator, that is, a function u : Br0 → ℝ such that Lu = δ0. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of |x|, plus homogeneous polynomials multiplied by log(|x|) if the dimension n is even) which improves the classical result of [6]. The control we have on the complexity of each homogeneous term is optimal and in particular, when L is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to Kodaira [8]. The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by ℤ². [ABSTRACT FROM AUTHOR]

The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the mirrors of the group's reflections. The symmetric group version of this is known as the Calogero-Moser model of N identical particles on a line. This paper focuses on the group of symmetries of the regular icosahedron, associated to the root system of type H3. Special wavefunctions are defined by a generating function arising from the vertices of the icosahedron and have the key property of allowing easy calculation of the effect of the Dunkl Laplacian. The ground state is the product of a Gaussian function with powers of linear functions coming from the root system. Two types of wavefunctions are considered, inhomogeneous polynomials with specified top-degree part, and homogeneous harmonic polynomials. The squared norms for both types are explicitly calculated. Symmetrization is applied to produce the invariant polynomials of both types, as well as their squared norms. The action of the angular momentum square on the harmonic homogeneous polynomials is determined. There is also a sixth-order operator commuting with the Hamiltonian and the group action. [ABSTRACT FROM AUTHOR]

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent. [ABSTRACT FROM AUTHOR]

D. Cvetković, M. Doob and H. Sachs considered the high order eigenvalue problems of graphs. The high order eigenvalues of a graph G are solutions of the high degree homogeneous polynomial equations derived from G. We propose the adjacency tensor A (G) of a graph G and show that the high order eigenvalues of G can be regarded as eigenvalues of A (G). Some results of the spectrum of the adjacency matrix are extended to the spectrum of A (G) by using the spectral theory of nonnegative tensors. An upper bound of chromatic number is given via the spectral radius of A (G). Our upper bound is a generalization of Wilf's bound χ (G) ≤ ρ (G) + 1 (where ρ (G) is the spectral radius of the adjacency matrix of a graph G) and sharper than the bound of Wilf in some classes of graphs. A formula of the number of cliques of fixed size which involve the spectrum of A (G) is obtained. [ABSTRACT FROM AUTHOR]

Any collection of |$n$| compact convex planar sets |$K_1,\dots , K_n$| defines a vector of |${n\choose 2}$| mixed areas |${\operatorname {V}}(K_i,K_j)$| for |$1\leq i

This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We establish the existence of a simultaneous extension from a majorizing sublattice in the classes of regular multilinear operators and regular homogeneous polynomials on vector lattices. By simultaneous extension from a sublattice we mean a right inverse of the restriction operator to this sublattice which is an order continuous lattice homomorphism. The main theorems generalize some earlier results for orthogonally additive polynomials and bilinear operators. The proofs base on linearization by Fremlin's tensor product and the existence of a right inverse of an order continuous operator with Levy and Maharam property. [ABSTRACT FROM AUTHOR]

We find a formula, in terms of n,d$n, d$, and p$p$, for the value of the F$F$‐pure threshold for the generic homogeneous polynomial of degree d$d$ in n$n$ variables over an algebraically closed field of characteristic p$p$. We also show that in every characteristic p$p$ and for all d⩾4$d\geqslant 4$ not divisible by p$p$, there always exist reduced polynomials of degree d$d$ in k[x,y]$k[x, y]$ whose F$F$‐pure threshold is a truncation of the base p$p$ expansion of 2d$\frac{2}{d}$ at some place; in particular, there always exist reduced polynomials f$f$ whose F$F$‐pure threshold is strictly less than 2deg(f)$\frac{2}{\deg (f)}$. We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez‐Betancourt, Witt, and Zhang, as to whether a list of necessary restrictions they prove on the F$F$‐pure threshold of reduced forms are "minimal" for p≫0$p\gg 0$. On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base p$p$ expansion of 2d$\frac{2}{d}$ that are always F$F$‐pure thresholds for reduced forms of degree d$d$, and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations. [ABSTRACT FROM AUTHOR]

Let |$X$| be a quasi-projective variety over a number field, admitting (after passage to |$\mathbb {C}$|⁠) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of |$X$| are sparse : the number of such points of height |$\leq B$| grows slower than any positive power of |$B$|⁠. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions. [ABSTRACT FROM AUTHOR]

In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue |$1$| can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space. Further, we show a formula relating the Alexander module's Hodge numbers to those of finite covers of the base space, under some assumptions. Dedicated to the memory of Georgia Benkart [ABSTRACT FROM AUTHOR]

In this paper, we study the semilinear elliptic equation − Δ u + a (x) | u | p − 2 u − b (x) | u | q − 2 u = 0 on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N ≥ 1 , where 2 ≤ p < q < + ∞. We first prove the existence of positive solutions to the above equation with constant coefficients a ¯ , b ¯. Then we establish a decomposition of Palais-Smale sequences for the functional with variable coefficients a (x) , b (x) , which tend to the constants a ¯ , b ¯ at infinity. [ABSTRACT FROM AUTHOR]