Homogeneous Polynomial Systems Research Articles

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Multigraded Sylvester forms, duality and elimination matrices

In this paper we study the equations of the elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

Quantitative Singularity Theory for Random Polynomials

Abstract Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

Contributions to Modified Spherical Harmonics in Four Dimensions

A modification of the classical theory of spherical harmonics in four dimensions is presented. The space $$\mathbb {R}^4 = \{(x,y,t,s)\}$$ is replaced by the upper half space $${\mathbb {R}}_{+}^{4}=\left\{ (x,y,t,s), s > 0 \right\} $$ , and the unit sphere S in $$\mathbb {R}^4$$ by the unit half sphere $$S_{+}=\left\{ (x,y,t,s): x^2 +y^2+ t^2+ s^2 =1, s > 0 \right\} $$ . Instead of the Laplace equation $$\Delta h = 0$$ we shall consider the Weinstein equation $$s\Delta u + k \frac{\partial u }{\partial s}= 0$$ , for $$k \in \mathbb {N}$$ . The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on $$S_{+}$$ . It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct—with respect to this non-Euclidean scalar product—an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation.

Generalized minimum distance functions

Using commutative algebra methods, we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If $${\mathbb {X}}$$ is a set of projective points over a finite field and I is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I are equal to the rth generalized Hamming weight of the corresponding Reed–Muller-type code $$C_{\mathbb {X}}(d)$$ of degree d. We show that the generalized footprint function of I is a lower bound for the rth generalized Hamming weight of $$C_{\mathbb {X}}(d)$$ . Then, we present some applications to projective nested Cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then, we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine Cartesian code.

Green's function of Dirichlet problem for biharmonic equation in the ball

ABSTRACTAn explicit representation of the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given. Expansion of the constructed Green's function in the complete system of homogeneous harmonic polynomials that are orthogonal on the unit sphere is obtained. Solution of the homogeneous Dirichlet problem with a simple right-hand side of the biharmonic equation is found.

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

SummaryIn this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.

Stable Cohomology of Spaces of Non-Resultant Systems of Homogeneous Polynomials in ℝn

Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in ℝn are calculated.

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

In the two articles in Appl. Math. Comput., J. Giné [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Giné’s results with one more limit cycle for each case.

Stable cohomology of spaces of non-resultant polynomial systems in ℝ 3

The stabilization of cohomology rings of spaces of non-resultant homogeneous polynomial systems of growing degree in ℝ3 is studied. The rational stable cohomology rings are explicitly calculated, and the instant of stabilization is estimated.

Stabilization of homogeneous polynomial systems in the plane

In this paper, we study the problem of stabilization via homogeneous feedback of single-input homogeneous polynomial systems in the plane. We give a complete classification of systems for which there exists a homogeneous stabilizing feedback that is smooth on $\mathbb{R}^2 \setminus\{ (0,0)\}$ and preserve the homogeneity of the closed loop system. Our results are essentially based on Theorem of Hahn in which the author gives necessary and sufficient conditions of stability of homogeneous systems in the plane.

Solving polynomial systems in integers

The paper describes a method for determining integer solutions of a homogeneous polynomial system with integer coefficients which has finitely many solutions in the projective space over the field of complex numbers under the assumption that these solutions have a certain property.

Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in $${\mathbb R}^2$$ R 2 and $${\mathbb C}^2$$ C 2

The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials \({\mathbb R}^2 \rightarrow {\mathbb R}\), and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in \({\mathbb C}^2\).

Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in ℝ3

The naturally topologized order complex of proper algebraic subsets in RP2 defined by systems of quadratic forms has the rational homology of S13.

On quadratizations of homogeneous polynomial systems of ODEs

On quadratizations of homogeneous polynomial systems of ODEs

Number of limit cycles for homogeneous polynomial system

Number of limit cycles for homogeneous polynomial system

The scattering variety

The so-called Scattering Equations which govern the kinematics of the scattering of massless particles in arbitrary dimensions have recently been cast into a system of homogeneous polynomials. We study these as affine and projective geometries which we call Scattering Varieties by analyzing such properties as Hilbert series, Euler characteristic and singularities. Interestingly, we find structures such as affine Calabi-Yau threefolds as well as singular K3 and Fano varieties.

Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.

Computation of Lyapunov Quantities of Homogeneous Quartic Polynomial System

The analysis of system behaviour near boundary of the stability domain requires the computation of Lyapunov quantities which determine a system behaviour in the neighbourhood of boundary This paper presents the computation of Lyapunov quantities and focal values of a homogeneous quartic polynomial system by using the Lyapunov-Poincare method where the polynomials are of degree four. The proposed two main theorems were proved to accomplish this goal. Keywords: Lyapunov quantities; focal values; Lyapunov-Poincare method; quartic polynomial system. 2010 Mathematics Subject Classification: 34C07; 34D08

A new proof for the correctness of the F5 algorithm

In 2002, Faugere presented the famous F5 algorithm for computing Grobner basis where two cri- teria, syzygy criterion and rewritten criterion, were proposed to avoid redundant computations. He proved the correctness of the syzygy criterion, but the proof for the correctness of the rewritten criterion was left. Since then, F5 has been studied extensively. Some proofs for the correctness of F5 were proposed, but these proofs are valid only under some extra assumptions. In this paper, we give a proof for the correctness of F5B, an equivalent version of F5 in Buchberger's style. The proof is valid for both homogeneous and non-homogeneous polynomial systems. Since this proof does not depend on the computing order of the S-pairs, any strategy of selecting S-pairs could be used in F5B or F5. Furthermore, we propose a natural and non-incremental variant of F5 where two revised criteria can be used to remove almost all redundant S-pairs.