Showing papers on "Square-free polynomial published in 1981"

Polynomial filters of any degree

Abstract: The general analytic expression for the polynomial smoothing function of any degree for equally spaced data points is presented. In addition to the explicit formula, a simple recursion relation is also given. The determination of numerical coefficients in the convolution equation involves only integer arithmetic. These results are further used to describe in some detail the effectiveness of digital polynomial smoothing, or filtering, of sampled spectral data in their dependence on the degree K of the polynomial, the number S of smoothing passes, and the range T of points in the smoothing interval. Then it can be shown that the sharpness of the frequency cutoff increases with the degree of the polynomial, the high-frequency attenuation increases with the number of smooths, and the cutoff of the filter moves toward lower frequencies as the range of points in the smoothing interval increases. The values of these three parameters should not be chosen entirely independently of one another, but the first two should be selected before the third.

On polynomial factorization over finite fields

Abstract: Let f(x) be a polynomial over a finite field F. An algorithm for determining the degrees of the factors of f(x) is presented. As in the Berlekamp algorithm (1968) for determining the factors of f(x), the Frobenius endomorphism on F[x]/(f(x)) plays a central role. Little-known theorems of Schwarz (1956) and Cesaro (1888) provide the basis for the algorithm we present. New and stream-lined proofs of both theorems are provided.

A note on complete sets for the polynomial-time hierarchy

Abstract: The polynomial-time hierarchy was introduced for the classification of problems that are probably more complex than those in NP. We recall some notations that may be found in greater detail in [2,3,4,53. The language accepted by an oracle machine M with oracle set A will be denoted by M(A). Define NP(A) = M(A) : M is a nondeterministic oracle machine that operates in time p(n) where p is some polynomial}. For a class of sets C, NP(C) = ~{NP(A) : A GC} . The classes ~ of the polynomial-time hierarchy are defined as follows.

On the structure of maximal (A, B) -Invariant subspaces: A polynomial matrix approach

Abstract: Given a controllable and observable triple ( A, B, C ) describing a linear time invariant multivariable system Σ, which gives rise to a full rank transfer function matrix T_{o}(s) , the structure of the maximal ( A, B )- invariant subspace contained in \ker C is investigated using a polynomial matrix approach. Thus, certain connections between the geometric and the polynomial matrix approaches to linear system theory are established.

New generating functions for certain polynomial systems associated with the H-functions

Abstract: This paper presents a number of new generating functions for several interesting classes of polynomials whose coefficients involve the //-function of C. Fox [Trans. Amer. Math. Soc. 98 (1961), 395-429]. The first of these main results stems from an attempt to provide a generalization of certain bilateral generating functions, due to H. M. Srivastava and R. Panda [J. Reine Angew. Math. 283/284 (1976), 265-274] and R. K. Raina [Proc. Nat. Acad. Sci. India Sect. A 46 (1976), 300-304], for a general class of hypergeometric polynomials or polynomials with essentially arbitrary coefficients; the other results are natural further generalizations of (or motivated by) the first one. It is observed how readily these new generating functions can be extended to hold for the H-function of several complex variables, which was defined in the aforementioned paper by Srivastava and Panda.

Solution of polynomial equations over the complex field

Abstract: Finding all n roots of the general polynomial over the complex field can be difficult theoretically. Therefore, presented here is an approach that attempts to combine algebraic theory (factor theorem) with multi‐stage Monte Carlo optimization to find the n roots of the general polynomial. An example of degree twenty‐five is presented. Rounding error problems for polynomials of degree fifty or over are discussed also.

Polynomial functions over finite monoids

Abstract: This paper is a completion of [7]; we characterize the monoids (S,.,e) such that the algebra (P(n,S), on) of polynomial functions over S is congruence free.