Showing papers on "Square-free polynomial published in 1996"

Arithmetic and factorization of polynomial over F2 (extended abstract)

Abstract: We describe algorithms for polynomial multiplication and polynomial factorization over the binary field IF2.and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one dqy of CPU time.

A transformation for 2-D systems and its invariants

Abstract: Zero coprime equivalence of 2-D polynomial matrices is considered and its connection with 2-D unimodular equivalence established. Certain invariants are determined.

Parallel polynomial operations on SMPs: an overview

Abstract: SMP-based parallel algorithms and implementationsfor polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p -adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.

A Synthesis of Certain Polynomial Sequences

Abstract: Encouraged by the comments of the reviewer [1] of my earlier article [4], I now take the opportunity to extend the material in [4] to incorporate some new thoughts on general recursively-defined polynomial sequences of the second order.

Lower bounds for polynomial evaluation and interpolation problems

Abstract: We show that there is a set of pointsp 1,p 2,...,p n such that any arithmetic circuit of depthd for polynomial evaluation (or interpolation) at these points has size $$\Omega \left( {\frac{{n\log n}}{{\log (2 + d/\log n}}} \right).$$ Moreover, for circuits of sub-logarithmic depthd, we obtain a lower bound of Ω(dn 1+1/d ) on its size.

Random Polynomials and Polynomial Factorization

Abstract: We give a precise average-case analysis of a complete polynomial factorization chain over finite fields by methods based on generating functions and singularity analysis.

Solving nonlinear recursions

Abstract: A general method to map a polynomial recursion on a matrix linear one is suggested. The solution of the recursion is represented as a product of a matrix multiplied by the vector of initial values. This matrix is product of transfer matrices whose elements depend only on the polynomial and not on the initial conditions. The method is valid for systems of polynomial recursions and for polynomial recursions of arbitrary order. The only restriction on these recurrent relations is that the highest‐order term can be written in explicit form as a function of the lower‐order terms (existence of a normal form). A continuous analog of this method is described as well.

Multidimensional structured matrices and polynomial systems

Abstract: We apply and extend some well-known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing, that is, computations with structured matrices and solving a system of polynomial equations. In the first part of our paper, we extend the Toeplitz and Hankel structures of matrices and some of their known properties to some new classes of structured (quasi-Hankel and quasi-Toeplitz) matrices, naturally associated to systems of multivariate polynomial equations. In the second part of the paper, we prove some relations between these structured matrices, which extend the classical relations of the univariate case.

The number of irreducible factors of a polynomial, II

Abstract: Given a polynomial $f\in k[x]$, $k$ a number field, we consider bounds on the number of cyclotomic factors of $f$ appropriate when the number of non-zero coefficients of the polynomial, $N(f)$, is substantially less than than its degree. In particular we obtain bounds which (apart from a small degree dependence) are only polynomial in $N(f)$. These results arise from variants of Mann's theorem on linear relations between roots of unity.

Maximal ideals of skew polynomial rings of automorphism type

Abstract: This paper is concerned with the problem of description of maximal ideals in a skew polynomial ring R[t,S].

Efficient Deterministic Algorithms for Embedding Graphs on Books

Abstract: We derive deterministic polynomial time algorithms for book embedding of a graph G = (V, E), ¦V¦ = n and ¦E¦ = m. In particular, we present the first deterministic polynomial time algorithm to embed any bipartite graph in \(O(\sqrt m )\)pages. We then use this algorithm to embed, in polynomial time, any graph G in \(O(\sqrt {\delta ^ * (G) \cdot m} )\)pages, where δ*(G) is the largest minimum degree over all subgraphs of G. Our algorithms are obtained by derandomizing the probabilistic proofs.

The field of Nash functions and factorization of polynomials

Abstract: The algebraically closed field of Nash functions is introduced. It is shown that this field is an algebraic closure of the field of rational functions in several variables. We give conditions for the irreducibility of polynomials with Nash coefficients, a description of factors of a polynomial over the field of Nash functions and a theorem on continuity of factors. Introduction. A holomorphic function f in an open connected set Ω⊂ C is called a Nash function if there exists an irreducible polynomial P : C×C→ C such that P (λ, f(λ)) = 0 for λ ∈ Ω. In Section 2 of the paper, the field of Nash functions is introduced (Prop. 2.1). It is an algebraic closure of the field of rational functions (Thm. 2.4 and Cor. 2.5). In the literature it is shown that such a closure is embedded in the field of Puiseux power series ([W], Thm. 3.1, Ch. IV). In the definition of the Nash field, the main trouble is to construct a family of sets {ΩP } with appropriate properties (Thm. 1.1). The main problem is to obtain the simple connectedness of these sets, which is the key fact in the proof of the algebraic closedness of the Nash field. In Section 3, conditions of the irreducibility of polynomials with coefficients in the Nash field are given (Thm. 3.2) and, as a corollary, a generalization of the Krull Theorem in the complex domain (Cor. 3.3). In Section 5, a theorem on the continuity of factors of a decomposition of a polynomial as a function of parameters is given (Thm. 5.1). In the proof of this theorem, the key role is played by an effective interpretation of the well-known fact that an irreducible polynomial which is reducible over the algebraic closure of the field of its coefficients is a product of conjugate polynomials (Thm. 4.4). Here, systems of coefficients of conjugate polynomials form a cycle of Nash mappings. 1991 Mathematics Subject Classification: Primary 12D99; Secondary 12F99.

A Polynomial Taking Integer Values

Abstract: In [2] Sury proves that for integers a, >j 2 1(ai aj) (i-j) is also an integer. (The result follows immediately from the theory of Lie groups; the number turns out to be the dimension of an irreducible representation of SU(n).) Sury gives an elementalry but indirect proof, based on the stronger result that H__ 2 i > j 2 (X1i-ai 1) (Xe-j 1) E Z[ X ]. A direct proof of the original result can be deduced from properties of the Vandermonde determinant and the fact that binomial coefficients are integral. If we define A(a1, a2, a,,) = rHn > > j 2(a aj), then our task is to show that A(1, 2,...,n) = = 1)! divides A(a1, a2,..., a,) whenever a1 0. It is well known that A(a1, a2, a,,) is the value of the Vandermnonde deternninant

On the factorization of polynomial matrices over the domain of principal ideals

Abstract: We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.

An extension of the root perturbation $m$-dimensional polynomial factorization method

Abstract: In this paper, an extension of an m-D (multidimensional or multivariable) polynomial factorization method is investigated. The method is the \"root perturbation method\" which is recently proposed by the author. According to this method, one sets to zero all complex variables, except one variable, and factorizes the 1-D polynomial. Furthermore, the values of these variables vary properly. In this way, one can \"built\" the m-dimensional polynomial in its factorized form. However, in the \"root perturbation method\", an assumption is that the 1-D polynomial must have discrete roots. In this paper, a solution is given in the case that the 1-D polynomial may have multiple roots. This is achieved by a proper transformation of the complex variables. The present method is summarized by way of algorithm. A numerical (3-D) example is presented.