Showing papers on "Square-free polynomial published in 2005"

Deterministic polynomial identity testing in non-commutative models

Abstract: We give a deterministic polynomial time algorithm for polynomial identity testing in the following two cases: Non-commutative arithmetic formulas: The algorithm gets as an input an arithmetic formula in the non-commuting variables x1,···,xn and determines whether or not the output of the formula is identically 0 (as a formal expression) Pure arithmetic circuits: The algorithm gets as an input a pure set-multilinear arithmetic circuit (as defined by Nisan and Wigderson) in the variables x1,···,xn and determines whether or not the output of the circuit is identically 0 (as a formal expression). One application is a deterministic polynomial time identity testing for set-multilinear arithmetic circuits of depth 3. We also give a deterministic polynomial time identity testing algorithm for non-commutative algebraic branching programs as defined by Nisan. Finally, we obtain an exponential lower bound for the size of pure setmultilinear arithmetic circuits for the permanent and for the determinant. (Only lower bounds for the depth of pure circuits were previously known.)

Polynomial time quantum algorithm for the computation of the unit group of a number field

Abstract: We present a quantum algorithm for the computation of the irrational period lattice of a function on Zn which is periodic in a relaxed sense. This algorithm is applied to compute the unit group of finite extensions of Q. Execution time for fixed field degree over Q is polynomial in the discriminant of the field. Our algorithms generalize and improve upon Hallgren's work [9] for the one-dimensional case corresponding to real-quadratic fields.

Four lectures on polynomial absolute factorization

Abstract: Polynomial factorization is one of the main chapters of Computer Algebra. Recently, significant progress was made on absolute factorization (i.e., over the complex field) of a multivariate polynomial with rational coefficients, with two families of algorithms proposing two different strategies of computation. One is represented by Gao’s algorithm and is explained in Lecture 2. The other is represented by the Galligo-Rupprecht-Cheze algorithm, presented in Lectures 4 and 5. The latter relies on an original use of the monodromy map attached to a generic projection of a plane curve on a line. It also involves zero-sums relations (introduced by Sasaki and his collaborators) with efficient semi-numerical computations to produce a certified exact result.

The Cayley-Menger determinant is irreducible for n ≥ 3

Abstract: We prove that the Cayley-Menger determinant of an n-dimensional simplex is an absolutely irreducible polynomial for n ≥ 3. We also study the irreducibility of the polynomials associated to the related geometric constructions.

Polynomial interpretations as a basis for termination analysis of logic programs

Abstract: This paper introduces a new technique for termination analysis of definite logic programs based on polynomial interpretations. The principle of this technique is to map each function and predicate symbol to a polynomial over some domain of natural numbers, like it has been done in proving termination of term rewriting systems. Such polynomial interpretations can be seen as a direct generalisation of the traditional techniques in termination analysis of LPs, where (semi-) linear norms and level mappings are used. Our extension generalises these to arbitrary polynomials. We extend a number of standard concepts and results on termination analysis to the context of polynomial interpretations. We propose a constraint based approach for automatically generating polynomial interpretations that satisfy termination conditions.

On minor prime factorizations for n-D polynomial matrices

Abstract: A tractable criterion is presented for the existence of minor prime factorizations for a class of multidimensional (n-D) (n>2) polynomial matrices whose reduced minors and greatest common divisors have some common zeros. We also present a constructive method for carrying out the minor prime factorizations when they exist. The proposed method is further extended to a larger class of n-D polynomial matrices by an invertible variable transformation. Three illustrative examples are given to show the effectiveness of the proposed method.

Polynomial separation of points in algebras

Abstract: We show that for a wide variety of domains, including all Dedekind rings with finite residue fields, it is possible to separate any two algebraic elements a, b of an algebra over the quotient field by integer-valued polynomials (i.e. to map a and b to 0 and 1, respectively, with a polynomial in K[x] that maps every element of D to an element of D), provided only that the minimal polynomials of a and b in K[x] are co-prime (which is obviously necessary). In contrast to this, it is impossible to separate a, b ∈ D by a n × n-integermatrix-valued polynomial (a polynomial in K[x] that maps every n×n matrix over D to a matrix with entries inD), except in the trivial case where a−b is a unit ofD. (This is despite the fact that the ring of n× n-integer-matrix-valued polynomials for any fixed n is non-trivial whenever the ring of integer-valued polynomials is non-trivial.) 2000 Math. Subj. Classification: Primary 13F20; Secondary 13B25, 11C08, 15A36, 16B99.

On the Number of Trace-One Elements in Polynomial Bases for $$\mathbb{F}_{2^n}$$

Abstract: This paper investigates the number of trace-one elements in a polynomial basis for $$\mathbb{F}_{2^n}$$ . A polynomial basis with a small number of trace-one elements is desirable because it results in an efficient and low cost implementation of the trace function. We focus on the case where the reduction polynomial is a trinomial or a pentanomial, in which case field multiplication can also be efficiently implemented.

On weighted polynomial approximation with gaps

Abstract: Let $\alpha$ be a nonnegative continuous function on $\mathbb{R}$. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in $C_{\alpha}$, where $C_{\alpha}$ is the weighted Banach space of complex continuous functions $f$ on $\mathbb{R}$ with $f(t) \exp (-\alpha (t))$ vanishing at infinity.

Polynomial time algorithm for computing the top Betti numbers of semi-algebraic sets defined by quadratic inequalities

Abstract: For any fixed l > 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≥ 0,...,Ps ≥ 0, where each Pi ∈ R[X1,...,Xk] has degree ≤ 2, and computes the top l Betti numbers of S, bk-1(S), ..., bk-l(S), in polynomial time. The complexity of the algorithm, stated more precisely, is Σi=0l+2 (si k2O((l,s)). For fixed l, the complexity of the algorithm can be expressed as sl+2 k2O(l), which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in Rk defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain by letting l = k, an algorithm for computing all the Betti numbers of S whose complexity is k2O(s)

Fractional polynomial response surface models

Abstract: Second-order polynomial models have been used extensively to approximate the relationship between a response variable and several continuous factors. However, sometimes polynomial models do not adequately describe the important features of the response surface. This article describes the use of fractional polynomial models. It is shown how the models can be fitted, an appropriate model selected, and inference conducted. Polynomial and fractional polynomial models are fitted to two published datasets, illustrating that sometimes the fractional polynomial can give as good a fit to the data and much more plausible behavior between the design points than the polynomial model.

The higher-order matching polynomial of a graph

Abstract: Given a graph G with n vertices, let p(G,j) denote the number of ways j mutually nonincident edges can be selected in G. The polynomial M(x)=∑j=0[n/2](−1)jp(G,j)xn−2j, called the matching polynomial of G, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of length t, denoted by pt(G,j). We compare this higher-order matching polynomial with the usual one, establishing similarities and differences. Some interesting examples are given. Finally, connections between our generalized matching polynomial and hypergeometric functions are found.

Construction of characteristic polynomial of reciprocal graphs from the number of pendant vertices

Abstract: A method for construction of the characteristic polynomial (CP) coefficients of the three classes of reciprocal graphs, viz., Ln + n(p), Cn + n(p), and K1,n−1 + n(p), has been developed that requires only the value of n. The working formulas have been expressed in matrix product form, computer programs for which can easily be developed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Autonomous subbehaviours and output nulling subspaces

Abstract: The aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours. Specifically, given a behaviour, we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems. We use this to analyse the different module structures on some vectorial polynomial spaces. Subsequently, we apply this analysis to the study and the polynomial characterization of output nulling subspaces, controlled invariant subspaces and the natural, feedback induced, -module structures in these spaces. Toeplitz operators and Wiener-Hopf factorizations play an important role.

Polynomial Identity Testing for Depth 3 Circuits

Abstract: We study the identity testing problem for depth 3 arithmetic circuits ( $$\sum\prod\sum$$ circuit). We give the first deterministic polynomial time identity test for $$\sum\prod\sum$$ circuits with bounded top fanin. We also show that the rank of a minimal and simple $$\sum\prod\sum$$ circuit with bounded top fanin, computing zero, can be unbounded. These results answer the open questions posed by Klivans---Spielman (STOC 2001) and Dvir---Shpilka (STOC 2005).

Resolution of multiple roots of nonlinear polynomial systems

Abstract: In this paper we discuss the roots and multiplicities of univariate and bivariate nonlinear polynomial systems and present methods to compute them robustly. For univariate polynomial systems, we propose an algorithm called the Topological Degree Bisection (TDB) algorithm which is developed based on the concept of the topological degree of a certain Gauss map which is deduced from input polynomials. The algorithm subdivides an input domain and computes the topological degree for each subdivided domain, which provides information on root existence inside the domain and the multiplicities. This process continues until the size of the subdivided regions which contain roots is less than a certain tolerance. In the bivariate polynomial system case, we use a combination of resultants and the TDB algorithm to develop a procedure for locating the roots and computing their multiplicities. Our methods are robust and global in nature. The proposed methods are compared with a subdivision method for root finding, the Interval Projected Polyhedron (IPP) algorithm and applied for the improvement of the IPP algorithm. Complexity analysis of the proposed methods is discussed with examples which illustrate our techniques.

Structural Stability of Planar Polynomial Foliations

Abstract: We discuss natural notions of structural stability of planar polynomial foliations of fixed degree with respect to perturbation within the same restricted set, within the set of all polynomial vector fields of the same degree, and within the set of smooth vector fields. Characterization theorems for structural stability in the latter two settings are obtained as immediate corollaries of known results. We provide sufficient conditions and separate necessary conditions for structural stability of planar polynomial foliations with respect to perturbation within the set of planar polynomial foliations of the same degree.

An inverse spectral problem for differential systems on the half-line with multiplied roots of the characteristic polynomial

Abstract: An inverse problem of spectral analysis is studied for non-selfadjoint singular systems of ordinary differential equations on the half-line in the case of multiplied roots of the characteristic polynomial. We give a formulation of the inverse problem, study properties of spectral characteristics, and prove the uniqueness theorem for the solution of the inverse problem.

Graph clustering using heat content invariants

Abstract: In this paper, we investigate the use of invariants derived from the heat kernel as a means of clustering graphs. We turn to the heat-content, i.e. the sum of the elements of the heat kernel. The heat content can be expanded as a polynomial in time, and the co-efficients of the polynomial are known to be permutation invariants. We demonstrate how the polynomial co-efficients can be computed from the Laplacian eigensystem. Graph-clustering is performed by applying principal components analysis to vectors constructed from the polynomial co-efficients. We experiment with the resulting algorithm on the COIL database, where it is demonstrated to outperform the use of Laplacian eigenvalues.

Polynomial Bases for Continuous Function Spaces

Abstract: Let S ⊂ \(\mathbb{R}\) denote a compact set with infinite cardinality and C(S) the set of real continuous functions on S. We investigate the problem of polynomial and orthogonal polynomial bases of C(S).

Couniform dimension over skew polynomial rings

Abstract: In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M R to obtain results on the couniform dimension of the polynomial modules M[x], M[x −1], and M[x, x −1] over suitable skew extension rings.

Polynomial functions on the classical projective spaces

Abstract: The polynomial functions on a projective space over a eld K = R, C orH come from the corresponding sphere via the Hopf bration. The main theorem states that every polynomial function (x) of degree d is a linear combination of \elementary" functionsjhx;ij d .

Uniformly isochronous polynomial centers

Abstract: We study a specific family of uniformly isochronous polynomial systems. Our results permit us to solve a problem about centers of such systems. We consider the composition conjecture for uniformly isochronous polynomial systems.