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Let fn(x,y,z) be a symmetric homogeneous polynomial of degree n . In this paper, we give the necessary and sufficient conditions to have fn(x,y,z) 0 for n 6 and any nonnegative real numbers x,y,z . In addition, we extend some results to n = 7 and n = 8 , and then apply the proposed method to prove several elegant symmetric homogeneous polynomial inequalities of degree n , 4 n 8 . Mathematics subject classification (2010): 26D05.

We prove that, in the case of ground field of characteristic zero, the variety cited in the title has almost polynomial growth. We construct an algebra generating this variety and completely describe the structure of the multilinear part of the variety as a module of the symmetric group.

A Grobner-Shirshov basis (a combinatorial system of generators) is defined in the set of multilinear elements of a T-ideal of the free associative algebra with a countable set of indeterminates. A combinatorial version of the well-known Specht problem about the finite basedness of polynomial identities of an arbitrary associative algebra is formulated. A “combinatorial Spechtness” property of the multilinear product of commutators of degree 2 and the same property for the three-linear commutator are established.

We compare three levels of algebraic certificates for evaluating the maximum mod- ulus of a complex analytic polynomial, on a compact semi-algebraic set. They are obtained as translations of some recently discovered inequalities in operator theory. Although they can be stated in purely algebraic terms, the only known proofs for these decompositions have a transcen- dental character.

Agraïments: The first author is partially supported by NSFC-11271027 and NSFC- 11171267. First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most 1 limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with 1 limit cycles.

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x' = 1, y' = a(x)y2 +b(x)y +c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

In this paper we find necessary and suficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems. The first and second authors are partially supported by a MICINN/ FEDER grant number MTM2011-22877 and by a AGAUR (General- itat de Catalunya) grant number 2009SGR 381. The third author is partially supported by a MICINN/ FEDER grant number MTM2008- 03437, by a AGAUR grant number 2009SGR 410 and by ICREA Academia

In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209–212) for the proof of the conjecture on the maximum number of invariant straight lines of cubic and quartic polynomial differential systems; and also prove the conjecture in a previous paper of the second author (Qualitative theory of polynomial differential systems, Shanghai Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain special case of the n n degree polynomial systems. Furthermore, we will prove that cubic and quartic differential systems have invariant straight lines along at most six and nine different directions, respectively, and also show that the maximum number of the directions can be obtained.

This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×

Let R R be a prime algebra over a commutative ring K K with unity and let f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) be a multilinear polynomial over K K . Suppose that d d is a nonzero derivation on R R such that for all r 1 , … , r n r_{1}, \ldots , r_{n} in some nonzero ideal I I of R R , [ d ( f ( r 1 , … , r n ) ) , f ( r 1 , … , r n ) ] k = 0 \Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0 with k k fixed. Then f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) is central–valued on R R except when char R = 2 R=2 and R R satisfies the standard identity s 4 s_{4} in 4 variables.

In this paper we investigate whether a polynomial algebra can be realized as a cohomology ring of a topological space. Our main results are that we can split the realizable polynomial algebra into a tensor product of certain simple factors and that these factors are given explicitly whenp>7. What is worth mentioning is that most of these factors are known to be realizable.

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

We study algebraic surfaces in three-dimensional Euclidean space given explicitly by a homogeneous polynomial. We ascertain necessary and sufficient conditions for the Gaussian curvature to be of constant sign. We give an estimate on the absolute curvature of the surface in terms of the number of distinct linear factors occurring in the decomposition of the polynomial.

One considers the problem of the factorization of polynomial matrices over an arbitrary field in connection with their reducibility by semiscalar equivalent transformations to triangular form with the invariant factors along the principal diagonal. In particular, one establishes a criterion for the representability of a polynomial matrix in the form of a product of factors (the first of which is unital), the product of the canonical diagonal forms of which is equal to the canonical diagonal form of the given matrix. There is given also a method for the construction of such factorizations.

It is well known that the ring of polynomial invariants of a finite matrix group {\small $G$} becomes more and more messy the further one moves away from groups generated by pseudoreflections. Also, the number of generators has a tendency to become large. In this experimental study, we try to create evidence for the observation that these unpleasant properties might improve if one enlarges the ring of invariants in one way or another. For instance, {\small $G$} might fix a symplectic form. It can be used to turn the ring of invariants into a Lie algebra by introducing a {\small $G$}-invariant Poisson bracket. In the absence of an invariant symplectic form, one might still consider the Lie algebra of invariant polynomial vector fields simultaneously with the ring of polynomial invariants. If one is prepared to leave the realm of commutative rings and Lie algebras, one can also take {\small $G$}-invariant differential operators into account. Under these additional operations the number of generators necessary to create all invariants is often drastically decreased. A particularly nice situation arises if the group fixes a quadratic form and a symplectic form at the same time, because together they give rise to an endomorphism on the space of homogeneous invariants of any given degree that, for instance, can be used to single out more effective generators in the new sense. In the classical situation of fields of characteristic zero, to which this paper is restricted, the averaging operator is both theoretically and algorithmically an important tool, whose computational feasibility, however, decreases with increasing degrees. The methods presented here might also help to avoid this difficulty.

This paper shows that a graded polynomial algebra over F 2 {F_2} with Steenrod algebra action possesses an analog of the adjoint representation for the cohomology of the classifying space of a compact connected Lie group.

There are given, in certain cases, necessary and sufficient conditions for a truncated polynomial ring over a Z 2 {Z_2} -Poincaré algebra to again be a Poincaré algebra. Applications are a splitting theorem for Poincaré algebras and an algebraic bordism classification for real projective space bundles.

The main result is that A [ X ] A[X] , the polynomial ring in any number of indeterminates over a coherent ring A A of global dimension two, is coherent.

Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree 2 n + 1 2n + 1 which holds in the matrix ring F n ( n > 2 ) {F_n}(n > 2) is a consequence of the standard identity s 2 n {s_{2n}} . The notions of rigid and semirigid sequences of matrices are defined and treated.

1. The algebras A,. We discuss in this note the properties of certain polynomial algebras over an integral domain k, where k has characteristic 2. Special cases of these algebras arise in Algebraic Topology: first, as the mod 2 cohomology algebras of the classifying spaces for real vector space bundles; and secondly, as a certain subalgebra (defined by C. T. C. Wall, see [4]) of the Thom algebra of nonoriented differentiable manifolds [3]. Let r denote either a positive integer or oo. We set