Showing papers on "Square-free polynomial published in 1972"

On the zeros of a polynomial and its derivative

Abstract: then every zero of the derivative P'(z) is contained in the smallest convex set that contains 8. This theorem has been rather thoroughly investigated [2] and sharpened in several ways. However, there is one related question that deserves attention, namely given one specific zero z,b of P(z), what can be said about a neighborhood of zX that will always contain a zero of P'(z)? By a translation, followed by a stretching (or shrinking), we may require that all the zeros of P(z) are in 8, the closed unit disk. Further by a rotation we may set zS = a, where 0?< a ? 1. With this normalization the problem can now be formulated precisely. Let 6P(a, n) be the set of all nth degree polynomials P(z) that have all of their zeros in 8, and at least one zero at z = a, 0? a <1. Let D(a, n) be a minimal region with the property that if P(z) zP(a, n), then P'(z) has at least one zero in 5O(a, n). Describe 5O(a, n). This problem seems to be rather difficult. As far as the authors are aware, the first step in the direction of this problem is contained in the conjecture due to Iliev [1, p. 25]. Let g(a) be the intersection of the disk j z-a-aJ ?1 with 8. Then, according to lliev, g(a) always contains at least one zero of P'(z). In this paper we replace g(a) by a region C*(a) which is in general much smaller, and we conjecture that every P(z) in 6'(a, n), has at least one root in C*(a). We can prove this proposition if a= 1. The case a = 0 is trivial. For 0 a < 1, the conjecture is still open.

Quartics over (2ⁿ)

Abstract: A description of the factorization of a quartic polynomial over the field GF(2n) is given in terms of the roots of a related cubic.

A Contribution to the Problem of Weighted Polynomial Approximation

Abstract: is required. The weights for which B; (w;f)-+O for every w/EL_ satisfying (1. 6) were characterized in the papers of H. POLLARD [27], [28], N. I. ACillESER [1] and S. N. MERGELIAN [24]. For the weights W/l (P?!=O) we obtain B~P)(W/l;f)-+O for every /satisfying W/l/ELp if l~p<= resp. for every f satisfying W/l/EL_ and (1. 6), as a corollary of our main result. Conceming other investigations (which apply to much more general w, but only for more restricted c1asses of functionsf) we refer to M. M. DZRBASIAN [7], [8], A. S. DZAFAROV [6] and G. FREUD [14].

Regular greatest common divisor of two polynomial matrices

Abstract: Let T(λ) and V(λ) be two polynomial matrices having dimensions l x l and m x l respectively, with T(λ) regular and of degree n and V(λ) of degree at most n – 1. It has recently been shown that a necessary and sufficient condition for T and V to be relatively right prime is that a certain nlm x nl matrix R(T, V) have full rank. It is shown here that if T and V have a greatest common right divisor D(λ), then provided D is regular, its degree k is equal to n – (1/l) rank R. Furthermore, if R˜. denotes the matrix of the first (n – k) lm rows of R, then it is shown that the last (n – k) l columns of R˜ are linearly independent and that the coefficient matrices of D can be obtained by expressing the remaining columns of R˜ in terms of this basis.

Simple Proofs of Lower Bounds for Polynomial Evaluation

Abstract: The idea of establishing lower bounds on the number of arithmetic operations required to evaluate polynomials is due originally to Ostrowski (1954A). He showed that at least n multiplications and n additions/subtractions are required to evaluate nth degree polynomials for n ≦ 4. Since then, this result has been proved true for all nonnegative values of n [Belaga (1961A), Pan (1966A)]. Motzkin (1955A) introduced the idea of preconditioning; if the same polynomial is to be evaluated at many points, it may be reasonable to allow some free preprocessing of the coefficients. It has been shown [Motzkin (1955A), Belaga (1961A)] that even if this preconditioning is not counted then at least [n/2] multiplications/divisions and n additions/ subtractions are necessary to evaluate nth degree polynomials. Eve (1964A) and others have shown that this lower bound is almost achievable: an nth degree polynomial can be evaluated in [n/2] + 2 multiplications and n additions/subtractions, provided some irrational preconditioning is allowed without cost.

On the continuity of best polynomial approximations

Abstract: Supposef is a continuous complex valued function defined on a compact set E in the plane and pj(f, E) is the polynomial of degree n of best uniform approximation to f on E. If a polynomial qn of degree n approximates / on E "almost" as well as pn(f; E), then qn is "almost" pn(f; E). Sharp estimates, one for the real and one for the general case, are found for flqn-p,(f, E)W1E in terms of the quantity (jfJ-q EJ-lf'-p (f, E)j|4, where II J denotes the uniform norm on E.

Uniform Approximation of Functions Through Optimal Partitioning

Abstract: In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X in a Euclidean space is approached through the partitioning of X and the definition of a corresponding norm. If the width of this partition is sufficiently small, the unique best polynomial approximation in the corresponding norm is arbitrarily close to the set of best uniform polynomial approximations. Furthermore, for fixed k greater than the number of points in a critical point set, there exists a partition of X into k subsets, so that the corresponding best polynomial approximation is arbitrarily close to the set of best uniform polynomial approximations.

A note on generalized polynomial identities

Abstract: Let A be an algebra with 1 over a field F and let B be a fixed F-basis of A. Let F〈x〉=F〈x1,…, xn,…,〉 be the free algebra over F in noncommutative indeterminates x1,…, xn,…, and denote by A F 〈x〉 the free product of A and F〈x〉 over F. The elements of A F 〈x〉 of the form varies, repetitions allowed) form an F-basis of A F 〈x〉. They will be referred to as basis monomials, and the involved in a particular basis monomial will be called the coefficients of that basis monomial.