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In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Z n -graded polynomials in the nth q _ -Weyl algebra.The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Z n -graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring.The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples.

Let R be a unital associative algebra over a field K of characteristic zero, and let f be a multilinear polynomial of degree m over K. If m ≤ 3 , we prove that all traceless matrices can be written as the sum of two values of f evaluated over M n ( R ) with n ≥ 2 . If m = 4 , we prove the same result for n ≥ 3 .

In this note, we show that over fields of any characteristic, exponential sums of Boolean instantiations of polynomials computed by multilinear circuits can be computed by multilinear circuits with polynomial blow-up in size. In particular, multilinear-VNP equals multilinear-VP. Our result showing closure under exponential sums also holds for other restricted multilinear classes - polynomials computed by multilinear (bounded-width) algebraic branching programs and formulas. Furthermore, it holds even if the circuit class is not fully multilinear but computes a polynomial that is multilinear in the summation variables. We study exponential sums of polynomials in different models of computation.We show that polynomial-size multilinear circuits are closed under exponential sums.We need to use varying techniques based on the characteristic of the field.We show some essential differences between general circuits and multilinear circuits.

In this paper, we deal with two problems of static output feedback for input-affine polynomial dynamical systems. One is to design a static output-feedback controller so as to render a prescribed algebraic set invariant for the resulting closed-loop system. The other is to design a static outputfeedback controller so as to realize a prescribed vector field on a prescribed algebraic set. It is shown that the two problems can be represented by a particular inclusion of polynomials, and the inclusion can be solved. As a result, all the static output-feedback controllers required in the problems can be exactly represented by using free polynomial parameters.

Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models where the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) symmetric conjugate form. On the constraint side, the decision variables belong to one of the following three sets: (1) the $$m$$ m -th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogenous polynomial functions and establish key linkages between complex multilinear forms and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented. Numerical results are reported to illustrate the effectiveness of the proposed approximation algorithms.

Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices of trace $0$., Comment: 8 pages

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie–Yamaguti algebra. This identity is a consequence of the defining identities for Lie–Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a nonassociative polynomial with 8 terms. Our computations show that no such identities exist in degrees less than 8.

For let be the polynomial algebra in variables of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well-known rules. Let denote the image of the action of the positively graded part of A major problem is that of determining a basis for the quotient vector space Both and are graded where denotes the set of homogeneous polynomials of degree A spike of degree is a monomial of the form where for each In this paper we show that if and can be expressed in the form with then where is the number of spikes of degree

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.

The template-based method is one of the most successful approaches to algebraic invariant synthesis. In this method, an algorithm designates a template polynomial \(p\) over program variables, generates constraints for \(p=0\) to be an invariant, and solves the generated constraints. However, this approach often suffers from an increasing template size if the degree of a template polynomial is too high.

Agraïments: FEDER-UNAB10-4E-378 and Consolider CSD2007-00004 "ES" We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S - 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S - 1 algebraic limit cycles given by circles, and this number is reached.

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

It is described how set membership identification and model rejection for polynomial models can be described using polynomial inequalities and inequations. Using difference algebra methods these problems can be reduced form based on so called autoreduced sets. It is shown that these descriptions generalize state space descriptions. It is also discussed how special forms of autoreduced sets can make methods based on interval methods easier to implement.

A polynomial in three variables is considered near a singular point where the polynomial itself and its partial derivatives vanish. A method for calculating asymptotic expansions in parameters for all branches of the set of roots of the polynomial near the singular point is proposed. The method is based on spatial power geometry and uses modern computer algebra algorithms: for calculating Grobner bases and for work with algebraic curves. The implementation of the method is demonstrated on the example of a sixth-degree polynomial in three variables considered near infinity and near a degenerate singular point.

Agraïments: The third author is partially supported by FCT through CAMGDS, Lisbon. In this paper we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k = −2, −1, . . . , 3, 4 their polynomial integrability has been characterized. Here we have two main results. First we characterize the polynomail integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian system with homogeneous polynomial potentials. Finally we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.

It is shown that our description of the dynamics of interaction between atoms with microcavity mode quanta and an external electromagnetic field under conditions of two-photon resonance follows from a simple theory of one-photon resonance interaction between two-level atoms and the same external electromagnetic field, if we substitute su(2) algebra elements of the one-photon resonance theory for third-order elements of polynomial algebra. Commutation relations of polynomial algebra determine the features of processes with the participation of a single object, i.e. an atom-photon cluster localized in a microcavity. Its states are the basis for irreducible representations of this polynomial algebra.

Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algebra and the other one works directly in the Weyl algebra, thus requiring the use of tools from non-commutative computer algebra. In behavioral systems theory, the resulting model for the data is known as the most powerful unfalsified model (MPUM) within the class of linear systems with kernel representations over the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients.

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.

In this paper, we give some tools for the construction of homogeneous feedback which stabilizes homogeneous polynomial systems of odd degree in the plane.

Methods of modern algebra have appeared to be extremely useful in system theory and digital signal processing. New classes of linear filters and fast linear convolution algorithms have been developed based on the algebraic approach. In this paper, we apply this approach to the description of a class of polynomial (Volterra) filters of signals and fields defined over finite groups. Since the polynomial filters can be considered as a direct extension of linear filters, it is reasonable to apply the algebraic methods to a nonlinear case too. After a brief introduction to the abstract signal theory, a matrix representation of generalized polynomial filter is presented. Finally, we discuss the construction of fast nonlinear convolutions algorithms on the basis of their linear counterparts.

H-bases are bases for polynomial ideals, characterized by the fact that their homogeneous leading terms are a basis for the associated homogeneous ideal. In the computation ofH-bases without term orders, an important task is to determine the orthogonal projection of a homogeneous polynomial to certain subspaces of homogeneous polynomials with respect to a given inner product. One way of doing so is to use an orthogonal basis of the subspace. In this paper, we present and study a method to efficiently compute such a basis for a particular but important inner product.

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.

Let F be a field of characteristic zero. In this paper we study the variety of Leibniz algebras V ˜ 1 defined by the identity y 1 ( y 2 y 3 ) ( y 4 y 5 ) ≡ 0 . We give a complete description of the space of multilinear identities in the language of Young diagrams through the representation theory of the symmetric group. As an outcome we show that the variety V ˜ 1 has almost polynomial growth, i.e., the sequence of codimensions of V ˜ 1 cannot be bounded by any polynomial function but any proper subvariety of V ˜ 1 as polynomial growth.

We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on m-regularity for graded modules over polynomial algebras.

Commutative differentially simple rings have proved to be quite useful as a source of examples in noncommutative algebra. In this paper we use the theory of holomorphic foliations to construct new families of derivations with respect to which the polynomial ring over a field of characteristic zero is differentially simple.

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth q-Weyl algebra.The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Zn-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring.The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples.

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

We present three main results. The first two provide sufficient conditions in order that a planar polynomial vector field in C2 has a rational first integral, and the third one studies the number of multiple points that an invariant algebraic curve of degree n of a planar polynomial vector field of degree m can have in function of m and n.

This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' Gröbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gröbner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Approximate polynomial algebra becomes an emerging area of study in recent years with a broad spectrum of applications. In this paper, we present a software toolbox ApaTools for approximate polynomial algebra. This package includes Maple and Matlab functions implementing advanced numerical algorithms for practical applications, as well as basic utility routines that can be used as building blocks for developing other numerical and symbolic methods in computational algebra.

LetWbe a finite group acting on a latticeLover thep-adic integers Z ∧ p. The analysis of the ring of invariants of the associatedW-action on the algebra Z ∧ p[L] of polynomial functions onLis a classical question of invariant theory. Ifpis coprime to the order ofW, classical results show thatWis a pseudo reflection group, if and only if the ring of invariants is again polynomial. We analyze the situation for odd primes dividing the order ofWand, in particular, determine those pseudo reflection groups for which the ring of invariants Z ∧ p[L]Wis a polynomial algebra.

We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

The Mobius algebra of a poset was introduced by Solomon and also studied by Greene in the special case of lattices. Let L denote a geometric lattice. Our key idea is to consider all the characteristic polynomials of upper intervals in L as components of one object, which is multiplicative. Now, by simple algebraic means we obtain new identities involving the characteristic polynomial and the Tutte polynomial of a geometric lattice.

By using the results about polynomial matrix in the system and control theory, this paper gives some discussions about the algebraic properties of polynomial matrices. The obtained main results include that the ring of nxn polynomial matrices is a principal ideal, and principal one-sided ideal ring.

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 ]])×

We finished the preceding chapter with a notion of approximate zero of a function and an algorithmic scheme to compute these approximate zeros, the adaptive homotopy.

A classification is provided of two-dimensional homogeneous polynomial vector fields of arbitrary degree. This classification results from a consideration of the linear equivalence of such vector fields, using techniques from tensor and spinor algebra. It simplifies, synthesizes and generalizes previous works, which have been restricted to those cases when the polynomials are of no higher than third degree.

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.

We introduce the notion of invariant subspaces for multilinear operators from which the invariant subspace problems for multilinear and polynomial operators arise. We prove that polynomial operators acting in a finite dimensional complex space and even polynomial operators acting in a finite dimensional real space have eigenvalues. These results enable us to prove that polynomial and multilinear operators acting in a finite dimensional complex space, even polynomial and even multilinear operators acting in a finite dimensional real space have nontrivial invariant subspaces. Furthermore, we prove that odd polynomial operators acting in a finite dimensional real space either have eigenvalues or are homotopic to scalar operators; we then use this result to prove that odd polynomial and odd multilinear operators acting in a finite dimensional real space may or may not have invariant subspaces.

In this paper we present a polynomial process algebra (PPA) like basic process algebra which can be used to model both polynomial behavior of parallel systems. It provides a nature framework for the concurrent composition systems, and can deal with the nondeterministic behavior. This process algebra is obtained by the polynomial transition systems which we defined. In this paper we concentrate on giving the syntax and semantic, and meanwhile defining the bisimulation equivalence. In the last we give an example to illustrate it.

This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of multilinear ABPs to that of multilinear arithmetic formulas, and prove a tight super-polynomial separation between the two models. Specifically, we describe an explicit n-variate polynomial F that is computed by a linear-size multilinear ABP but every multilinear formula computing F must be of size nΩ(log n).