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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

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.

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.

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 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.

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

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.

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.

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.

It is proved that for a system of linear partial differential equations with polynomial coefficients, the Grobner basis in the Weyl algebra is sufficient for the computation of the characteristic variety. In particular, this yields a correct algorithm of computing the singular locus of a holonomic system with polynomial coefficients. The characteristic variety is defined analytically, i.e. by using the ring of power series, and it has not been obvious that it can be computed by purely algebraic procedure. Thus the algorithm of computing the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients can be readily implemented on a computer algebra system.

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.

We present new algorithms for exact multilinear k-monomial counting which is to compute the sum of coefficients of all degree-k multilinear monomials in a given polynomial P over a ring R described by an arithmetic circuit C. If the polynomial can be represented as a product of two polynomials with degree at most d < k, our algorithm can solve this problem in O*((n ↓d)) time, where (n ↓d) = Σi=0d (n i). O* omits a polynomial factor in n. For the general case, the proposed algorithm takes time O*((n ↓k) ). In both cases, our results are superior to previous approaches presented in [Koutis, I. and Williams, R.: Limits and applications of group algebras for parameterized problems. ICALP, pages 653-664 (2009)]. We also present a polynomial space algorithm with time bound O*(2k(n k)). By reducing the #k-path problem and the #m-set k-packing problem to the exact multilinear k-monomial counting problem, we give algorithms for these two problems that match the fastest known results presented in [2].

In this paper, we propose a model reduction scheme for a special class of polynomial dynamical systems. The biochemical processes and networks are our main motivation for this study. It is well known that many biochemical processes can be represented using quasi-polynomial systems. We show that a special class of quasi-polynomial systems can be cast in the Lotka-Volterra canonical form. For a given polynomial dynamical system, we propose a procedure to verify whether a given set of polynomials represents an invariant manifold of the system. Then, we study under what algebraic conditions a Lotka-Volterra system admits invariant manifolds. Finally, we combine our results with tools from differential algebra to propose a model reduction procedure for polynomial dynamical systems.

In this paper, we propose an efficient method to solve polynomial systems whose equations are left invariant by the action of a finite group G. The idea is to simultaneously compute a truncated SAGBI-Gr¨obner bases (a generalisation of Gr¨obner bases to ideals of subalgebras of polynomial ring) and a Gr¨obner basis in the invariant ring K[A1, . . . , An] where Ai is the i-th elementary symmetric polynomial. To this end, we provide two algorithms: first, from the F5 algorithm we can derive an efficient and easy to implement algorithm for computing truncated SAGBI-Gr¨obner bases of the ideals in invariant rings. A first implementation of this algorithm in C enable us to estimate the practical efficiency: for instance, it takes only 92s to compute a SAGBI basis of Cyclic 9 modulo a small prime. The second algorithm is inspired by the FGLM algorithm: from a truncated SAGBI-Gr¨obner basis of a zero-dimensional ideal we can compute efficiently a Gr¨obner basis in some invariant rings K[h1, . . . , hn]. Finally, we will show how this two algorithms can be combined to find the complex roots of such invariant polynomial systems.

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 .

This paper discusses the portability of SAC-2/ALDES and reviews some applications in polynomial algebra. Furthermore, we indicate that the concept of the safety variable in the SAC-2/ALDES is not proper. When, for example, we used safety variable in isolating complex roots of polynomials, something wrong happened.

An algebraic surface public-key cryptosystem was developed by Akiyama and Goto. Its security is based on a decision randomizing polynomial problem which is related to a problem of finding sections on fibered algebraic surfaces which can be reduced to solving a multivariate equation system known to be NP-complete. In the case that the defining equation of a surface used for public-key is in a certain form, Uchiyama and Tokunaga succeeded in attacking in the sense of getting plain texts from corresponding ciphertexts using reductions efficiently without solving section finding problem. In this paper, two algorithms applicable to all cases are suggested. One is the generalization of Uchiyama-Tokunaga's attack from polynomial ring over to polynomial ring over rational function field, and the other takes advantages of Grobner base techniques so as to deal with in the polynomial ring over .

Let I be a perfect height 2 homogeneous ideal in a graded polynomial algebra over a field. By a ‘liaison’ method we construct new examples of such ideals I with prescribed Hilbert function and number of generators in each degree.

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.

An efficient technique for solving polynomial systems with a particular structure is presented. This structure is very specific but arises naturally when computing the critical points of a symmetric polynomial energy function. This novel numerical solution method is based on homotopy continuation and is a particular application of results due to Canny and Rojas. An illustrative example from magnetism is presented.

In this paper, we consider the problem of reducing a multivariate polynomial matrix to Smith form by a unimodular equivalence transformation. Grobner bases are used to determine whether a certain class of multivariate polynomial matrices is equivalent with its Smith form. The proposed conditions can be easily tested using a computer algebra system.

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation., 26 pages. A Maple implementation is available

Robust stabilization of uncertain systems has been extensively investigated and the stability test for the whole set of uncertain parameters has been reduced to a finite number of test points, four points for the characteristic polynomial with independent coefficients. As a result the robust stability margin can be determined using a reasonable amount of computation. It is impossible to apply the results of the test to a practical system as the coefficients of the characteristic polynomial for a physical system are usually functions of uncertain parameters. However, many physical systems may be represented by a second-order mass-spring-damper system with a special multilinear form in its characteristic polynomial. This paper investigates second-order mass-spring-damper systems and the reduction of the number of test points. It is shown that such a system with arbritrary compensators always has a multilinear characteristic polynomial. It is also shown that a line in the two-dimensional parameter space forms the boundary after the mapping of a multilinear characteristic polynomial and this interior extreme line forms a conic curve in the complex plane. The boundary of uncertain domain for a multilinear polynomial with two uncertainty parameters can be determined analytically using this curve, and the four sides image of a square of the uncertain parameter. Therefore, the stability margin may be determined by checking the intersections of the boundary with the zero point. A similar procedure can be used for second-order systems with more than two uncertainty parameters when parameter optimization is used in determining the boundary.

The purpose of this note is to re-establish, with a new and simple proof, a theorem which in fact represents the main results of [1].

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.

It is proved that the algebra of the KdV‐invariant polynomial functionals on the space of C∞ functions on the one‐dimensional torus is isomorphic to the polynomial algebra of the infinite number of the conserved quantities found by Miura, Gardner, and Kruskal [J. Math. Phys. 9, 1204 (1968)]. As a corollary, it is shown that the algebra of the isospectral polynomial functionals of the Hill’s operator also coincides with this algebra.

This paper describes an application of the theory of polynomial models to the study of some natural objects in geometric control theory. In particular, it utilizes the correspondence between factorization of polynomial matrices and invariant subspaces to obtain, by the use of Toeplitz operators, a polynomial characterization of (A, B) invariant subspaces as well as those included in her C. A geometric characterization of feedback irreducibility is rederived.

A geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials. It is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues, quantities normally associated with a linear system. These quantities are then employed in studying stability properties. The necessary and sufficient conditions for all forms of stabilities characteristic of a two-dimensional system are provided. This result, together with the classical theorem of Frommer, completes a stability analysis for a two-dimensional homogeneous polynomial system.

Second order linear homogeneous ODEs which generalize those considered in the orthogonal polynomial theory are studied. A method and an algorithm to approach their polynomial solutions are also given.

This paper briefly presents the fundamental facts of the polynomial systems theory as applied by Rosenbrock, Wolovich and others. The theory utilizes an operator algebra which is a natural generalization of the usual operational calculus based on the Laplace and Z-transforms. It is shown how this operator algebra can be used in an efficient way for solving a number of typical problems relating to the analysis and synthesis of linear time-invariant feedback systems and estimators.

(1982). A particular integral of Polynomial form. International Journal of Mathematical Education in Science and Technology: Vol. 13, No. 3, pp. 359-370.