Homogeneous Polynomials Of Degree Research Articles

Bifurcation of limit cycles from a centre in ℝ4 in resonance 1:N

For every positive integer N ≥ 2 we consider the linear differential centre in ℝ4 with eigenvalues ±i and ±Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.

Perturbations of parallel flows on the sphere in [formula omitted

We consider perturbations of the parallel flows on each invariant sphere x 2 + y 2 + z 2 = h ( > 0 ) by homogeneous polynomials of degree m ( ⩾ 2 ) , and obtain the least upper bound of the number of limit cycles appearing on the sphere by using the Poincaré map and the averaging method.

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré–Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17–30] the Poincaré–Liapunov constants are calculated for polynomial systems of the form x ̇ = − y + P n ( x , y ) , y ̇ = x + Q n ( x , y ) , where P n and Q n are homogeneous polynomials of degree n . The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré–Liapunov constants, through the study of the highest fine focus order for n = 4 and n = 5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré–Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870–1877]. Moreover, the computational problems which appear in the computation of the Poincaré–Liapunov constants and the determination of the number of functionally independent ones are also discussed.

1:−3 resonant centers on C2 with homogeneous cubic nonlinearities

We characterize the collection of systems of differential equations on C 2 of the form x ̇ = x + p ( x , y ) , y ̇ = − 3 y + q ( x , y ) , where p and q are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of ( 0 , 0 ) of the form x 3 y + ⋯ , where omitted terms are of order at least five. Such systems are called 1 : − 3 resonant centers.

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphic partial differential operators whose principal part $Q_{2p}(D)$ essentially is the (complex) Laplace operator to a power, $\Delta^p$. We pose inital data on a singular conic divisor given by P=0, where $P$ is a homogeneous polynomial of degree $2p$. We show that this problem is uniquely solvable if the polynomial $P$ is elliptic, in a certain sense, with respect to the principal part $Q_{2p}(D)$.

Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z) , Z \in C^{n}$. We call a such polynomial $p$ H-Stable if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous H-Stable polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \min(deg_{p}(i),i)$ and $Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n} {p(x_1,...,x_n)\over x_1 \cdots x_n}$ then the following inequality holds $${\partial^n\over\partial x_1...\partial x_n} p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} \bigg({C_i-1\over C_i}\bigg)^{C_{i}-1}.$$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the H-Stable polynomials which are products of linear forms. Our proof is relatively simple and "noncomputational"; it uses just very basic properties of complex numbers and the AM/GM inequality.

On the Alexander–Hirschowitz theorem

The Alexander–Hirschowitz theorem says that a general collection of k double points in P n imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3 , where our proof is shorter. We end with an account of the history of the work on this problem.

Graphs and the Jacobian conjecture

It is proved in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proceedings of the AMS 133 (8) (2005) 2201–2205] that it suffices to study the Jacobian Conjecture for maps of the form x + ∇ f , where f is a homogeneous polynomial of degree d ( = 4 ) . The Jacobian Condition implies that f is a finite sum of d -th powers of linear forms, 〈 α , x 〉 d , where 〈 x , y 〉 = x t y and each α is an isotropic vector i.e. 〈 α , α 〉 = 0 . To a set { α 1 , … , α s } of isotropic vectors, we assign a graph and study its structure in case the corresponding polynomial f = ∑ 〈 α j , x 〉 d has a nilpotent Hessian. The main result of this article asserts that in the case dim ( [ α 1 , … , α s ] ) ≤ 2 or ≥ s − 2 , the Jacobian Conjecture holds for the maps x + ∇ f . In fact, we give a complete description of the graphs of such f ’s, whose Hessian is nilpotent. As an application of the result, we show that lines and cycles cannot appear as graphs of HN polynomials.

Linearizability of linear systems perturbed by fifth degree homogeneous polynomials

We present the necessary and sufficient conditions for linearizability of the planar complex system , where P and Q are homogeneous polynomials of degree 5. Using these conditions, we also give the complete solution for the isochronicity of real systems in the form of linear oscillator perturbed by fifth degree homogeneous polynomials.

A quartic system and a quintic system with fine focus of order 18

By using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems E n : z ˙ = i z + R n ( z , z ¯ ) , where R n is a homogeneous polynomial of degree n, in this note we construct two concrete examples, E 4 and E 5 , such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations: E 4 : z ˙ = i z + 2 i z 4 + i z z ¯ 3 + 52278 20723 e i θ z ¯ 4 , where θ ∉ { k π ± π 6 , k π + π 2 , k ∈ Z } , and E 5 : z ˙ = i z + 3 z 5 + 20 ( c + 3 ) 9 c 2 − 15 z 4 z ¯ + z z ¯ 4 + 20 ( c + 3 ) c 2 9 c 2 − 15 z ¯ 5 , where c is the root between ( − 3 , − 5 / 3 ) of the equation 4155 c 6 − 10716 c 5 − 63285 c 4 − 18070 c 3 + 168075 c 2 + 205450 c + 60375 = 0 .

NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with n < m and n is odd, m is even. The tool for proving these results is based on a method developed in [7].

Oscillatory integral operators with homogeneous polynomial phases in several variables

We obtain L 2 decay estimates in λ for oscillatory integral operators T λ whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of ( 2 + 2 ) -dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large. The proof for large m follows from essentially algebraic considerations. For cubics in ( 2 + 2 ) -dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the general approach of Phong and Stein [D.H. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994) 703–722].

Method of orbit sums in the theory of modular vector invariants

Let be a field, a finite-dimensional -vector space, a finite group, and the -fold direct sum with the diagonal action of . The group acts naturally on the symmetric graded algebra as a group of non-degenerate linear transformations of the variables. Let be the subalgebra of invariants of the polynomial algebra with respect to . A classical result of Noether [1] says that if , then is generated as an -algebra by homogeneous polynomials of degree at most , no matter how large can be. On the other hand, it was proved by Richman [2], [3] that this result does not hold when the characteristic of is positive and divides the order of . Let , , be a prime number, a finite field of elements, a linear -vector space of dimension , and a cyclic group of order generated by a matrix of a certain special form. In this paper we describe explicitly (Theorem 1) one complete set of generators of . After that, for an arbitrary complete set of generators of this algebra we find a lower bound for the highest degree of the generating elements of this algebra. This is a significant extension of the corresponding result of Campbell and Hughes [4] for the particular case of . As a consequence we show (Theorem 3) that if and is an arbitrary finite group, then each complete set of generators of contains an element of degree at least , where is a positive integer dependent on the structure of the generating matrix of the group . This result refines considerably the earlier lower bound obtained by Richman [3].

On primitives of monogenic functions†

Let Ω be a suitable open subset of Euclidean space , let Dx be the Cauchy–Riemann operator in and let M(Ω) be the space of left monogenic functions in Ω. Then it is shown that for each f ∈ M(Ω) there exists f ∈ M(Ω) such that , i.e. f admits a left monogenic primitive F in Ω. Furthermore, if for , M + (k) denotes the space of left monogenic homogeneous polynomials of degree k in , then for each Pk ∈ M + (k) a particular primitive P k+1 ∈ M + (k+1) of Pk is explicitly constructed. †Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

We consider the class of polynomial differential equations x˙ P n ( x,y)+ P n+1 ( x,y)+ P n+2 ( x,y), y˙= Q n ( x,y)+ Q n+1 ( x,y)+ Q n+2 ( x,y), for n ≥ 1 and where P i and Q i are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle.

Explicit non-algebraic limit cycles for polynomial systems

We consider a system of the form x ˙ = P n ( x , y ) + xR m ( x , y ) , y ˙ = Q n ( x , y ) + yR m ( x , y ) , where P n ( x , y ) , Q n ( x , y ) and R m ( x , y ) are homogeneous polynomials of degrees n, n and m, respectively, with n ⩽ m . We prove that this system has at most one limit cycle and that when it exists it can be explicitly found and given by quadratures. Then we study a particular case, with n = 3 and m = 4 . We prove that this quintic polynomial system has an explicit limit cycle which is not algebraic. To our knowledge, there are no such type of examples in the literature. The method that we introduce to prove that this limit cycle is not algebraic can be also used to detect algebraic solutions for other families of polynomial vector fields or for probing the absence of such type of solutions.

Decay rates of oscillatory integral operators in “1 + 2” dimensions

Consider the oscillatory integral operator , where is real valued, and λ is a large real number. We prove that, if S(x, y) is a homogeneous polynomial of degree n ≥3 satisfying a non-degeneracy assumption, then , and the decay rate is sharp (up to a power of log λ when n = 3).

An explicit representation as quasi-sum of squares of a polynomial generated by the AG inequality

An explicit representation of the difference (x1+· · ·+xn)ⁿ−nⁿx1· · · xn for all natural n ≥ 2 is given as a sum of pij(xi − xj)2 over all 1 ≤ i < j n where pij = pij(x1, . . . , xn) are homogeneous polynomials of degree n − 2 whose coefficients at all possible monomials of degree n − 2 are positive.

Polynomial-Time Computation of the Degree of a Dominant Morphism in Characteristic Zero. I

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables over the field of characteristic zero. We show how to compute the degree of a dominant rational morphism from W to W′. The morphism is given by homogeneous polynomials of degree d′.This algorithms is deterministic and polynomial in (dd′)n and the size of the input. Bibliography: 11 titles.

Polynomial projectors preserving homogeneous partial differential equations

A polynomial projector Π of degree d on H ( C n ) is said to preserve homogeneous partial differential equations ( HPDE) of degree k if for every f ∈ H ( C n ) and every homogeneous polynomial of degree k, q ( z ) = ∑ | α | = k a α z α , there holds the implication: q ( D ) f = 0 ⇒ q ( D ) Π ( f ) = 0 . We prove that a polynomial projector Π preserves HPDE of degree k , 1 ⩽ k ⩽ d , if and only if there are analytic functionals μ k , μ k + 1 , … , μ d ∈ H ′ ( C n ) with μ i ( 1 ) ≠ 0 , i = k , … , d , such that Π is represented in the following form Π ( f ) = ∑ | α | < k a α ( f ) u α + ∑ k ⩽ | α | ⩽ d D α μ | α | u α with some a α ’ s ∈ H ′ ( C n ) , | α | < k , where u α ( z ) ≔ z α / α ! . Moreover, we give an example of polynomial projectors preserving HPDE of degree k ( k ⩾ 1 ) without preserving HPDE of smaller degree. We also give a characterization of Abel–Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE.