Showing papers on "Square-free polynomial published in 1977"
TL;DR: In this paper, a Lagrange interpolation formula was proposed for the simultaneous determination of the zeros of real polynomials, which can be viewed as an improvement on that of Nourein.
44 citations
18 Jul 1977
TL;DR: NP-problems are considered in this paper as recognition problems over some alphabet Σ, i.e. A ⊂ Σ* is is an NP problem if there exists a NDTM (non-deterministic Turing machine) recognizing A in polynomial time.
Abstract: NP-problems are considered in this paper as recognition problems over some alphabet Σ, i.e. A ⊂ Σ* is is an NP problem if there exists a NDTM (non-deterministic Turing machine) recognizing A in polynomial time. It is easy to show that the following theorem holds true.
23 citations
TL;DR: In this article, the degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces Ng-' and two o-results and use pairs of inequalities of Bernsteinand Jackson-type to prove several direct and converse theorems.
Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces Ng-' (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernsteinand Jackson-type to prove several direct and converse theorems. For f in Ng-' we define a derivative D1 ?f in La, a = (n + p1)-1, which agrees with D f for smoothf, and prove scveral properties of D"?.
17 citations
01 Feb 1977
TL;DR: Theorem 3.2. as mentioned in this paper shows that projective C-modules are stably free, i.e., the C[y]-module P is locally extended from C by a slight generalisation of Quillen's theorem [10, Tlheorem 1] for some C-module Q.
Abstract: PROOF. Let S = k[x], C = D[x] and for each maximal ideal m of S, Cm = Sm ? O C. Let P be a finitely generated projective A-module with n = rank P > 2. Since KO(A) = KO(D), P is stably free. The maximal spectrum of Sm[y] is the union of two 1-dimensional sets [2, III, Proposition 3.13], hence by the cancellation theorem of Bass and Schanuel [2, IV, Theorem 3.1], Cm ?c P _ Cm[y]0. In other words, the C[y]-module P is locally extended from C. By a slight generalisation of Quillen's theorem [10, Tlheorem 1] P is extended from C, i.e. P _ C[y] %c Q for some C-module Q. Since projective C-modules are free, so is P. REMARK 1.2. Examples of nonfree projective ideals in D [x, y] are given in [8].
16 citations
30 Sep 1977
TL;DR: It is shown that the computation of the determinant requires an exponential number of multiplications if the commutativity of indeterminates is not allowed, which can reduce a computation of exponential complexity to a computations of polynomial complexity.
Abstract: In this paper we show that the computation of the determinant requires an exponential number of multiplications if the commutativity of indeterminates is not allowed. The determinant can be computed in polynomial time with the commutation of indeterminates. Hence the use of commutativity can reduce a computation of exponential complexity to a computation of polynomial complexity.
15 citations
9 citations
6 citations
01 Jan 1977
TL;DR: In this article, it is shown that a closer reexamination of Yun's 1976 paper reveals the reducibility of SQFR to GCD and the natural question that follows is whether GCD is reducible to SQFR.
Abstract: It is shown that a closer reexamination of Yun's 1976 paper reveals the reducibility of SQFR to GCD. The natural question that follows is whether GCD is reducible to SQFR. That is answered affirmatively and the derivation actually suggests an algorithm for computing GCD's when input polynomials are already represented by their SQFR form.
6 citations
3 citations
3 citations
TL;DR: In this paper, the algebraically closed dimension finite polynomial rings are determined by reducing the problem to two classes: skew polynomials and differential polynomorphisms.
Abstract: Let R be a noncommutative polynomial ring over the division ring K where K has center F. Then R = K[x,σ,D]where σ is a monomorphism of K and D is a σ-derivaton K. R is called dimension finite if (K: Fσ)<∞ and (K: FD)<∞ where Fσ is the subfield of F fixed under σand FD is the subfied of F of D-constants. R is algebraically closed if every nonconstant polynomial in Rfactors completely into linear factors. The algebraically closed dimension finite polynomial rings are determined. s done by reducing the problem to two classes: skew polynomial rings and differential polynomial rings. Examples algebraically closed polynomial rings which are not dimensfinite are given.
TL;DR: In this article, it was proved that the polynomials are all odd and that the integers are all O(n) odd and it is proved in [3] that 5
Abstract: Leeming [4] has defined a sequence of polynomials {Q 4n (x)} and a sequence of integers {Q 4n } by means of 1 and 2 Thus 3 Leeming showed that the Q 4n are all odd and that 4 It is proved in [3] that 5
01 Jan 1977
TL;DR: A generalization of program Dl (program D2) is presented which can be used to evaluate the first m derivatives of a polynomial if 4 does not necessarily divide n + 1 and how Ed must be computed to achieve a nearly optimal performance in this more general case.