Prime Polynomials Research Articles

Skew Prime多項式行列方程式の一解法

Skew Prime多項式行列方程式の一解法

One sided invertibility and localisation II

The aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).

Polynomial factorization in GF(2 m)

Polynomial factorization in the finite Galois field GF(2 m ) is the basis of the design of good error correcting codes. In this paper, an efficient algorithm for polynomial factorization in GF(2 m ) is proposed. Interesting properties of prime polynomial factors are deduced.

Rings, fields, the Chinese remainder theorem and an extension-Part II: applications to digital signal processing

For pt. I, see ibid., vol. 41, no. 10, p. 641-55 (1994). In Part I of the research work, we introduced an extension to the well known Chinese remainder theorem for processing polynomials with coefficients defined over a finite integer ring. We term this extension as the American-Indian-Chinese extension of the Chinese remainder theorem. A systematic procedure for factorizing a monic polynomial into pairwise relatively prime monic factor polynomials over integer rings was presented. This factorization is based on the corresponding factor polynomials, monic and relatively prime, over the associated finite field containing prime number of elements. In this paper, we study the application of the theory developed in Part I to deriving computationally efficient algorithms for performing tasks having multilinear form. Especially, we focus on the cyclic and acyclic convolution as they are two of the most frequently occurring computationally intensive tasks in digital signal processing. >

Polynomials generating Hamming codes

A class of polynomials generatingq-nary Hamming codes is studied. The criteria for a polynomial to belong to this class are established for the general case and for the case of prime polynomials. The conditions are determined under which reducible polynomials do not belong to the class of polynomials generating theq-nary Hamming codes.

One sided invertibility and localisation

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

A resultant criterion and formula for the inversion of a rational map in two variables

Generalizing the main theorem of Adjamagbo and van den Essen in their article “A resultant criterion and formula for the inversion of a polynomial map in two variables” (J. Pure Appl. Algebra 64 (1990) 1-6), we characterize in terms of resultant the birationality of a couple F of rational fractions in two variables over a field. From this characterization, we deduce first precise information about the degree of the couples of relatively prime polynomials which define the inverse of a birational F, then an inversion formula for a birational F, and finally two new algorithms for testing the birationality of F and computing its inverse when it exists.

SOME NEW RESULTS IN MULTIVARIABLE LINEAR SYSTEM: —Some Achievements in China, 1980–1984

From 1980—1984 some new results on multivariable linear system had been obtained by Chinese authors. The author of this paper gave out in 5 section the concept of (A, B) – characteristic subspace and its applications, the minimal polynomial matrix and its properties and the equivalence between state-space model and polynomial matrix and other corresponding results, some applications of Yokoyama canonical form especially in solving the disturbance decoupling problem, the properties of external skew prime polynomial matrices and their applications to output regulation, the internal model principle in frequence domain and robust control, and finally some other results were introduced.

A note on irregular prime polynomials in cyclotomic function field theory

Let r = p λ , K = F r ( t), f be an irreducible monic polynomial in F r [ t], K( Λ f ) the cyclotomic function field with conductor f, h f + the class number of the maximal “real” subfield of K( Λ f ), h f − the relative class number of K( Λ f ). The main results in this note are that (1) for each r = p λ > 2 there exist infinitely many irreducible monic polynomials f ∈ F r [ t] with p| h f −; (2) for each r = p λ there exist infinitely many irreducible monic polynomials f ∈ F r [ t] with p| h f +.

Fast Error Decoding with Binary VLSI Logic

Maximal distance binary codes that are composed of individual characters from the residues of pairwise prime polynomials are constructed and compared to Reed-Solomon codes. Although these binary residue codes are not as efficient as R-S codes in that codeword lengths are shorter, error decoding involves only binary and not finite field operations and thus allows faster decoding and greater data rates. Data rates of hundreds of megabits per second are feasible if decoding is implemented with VLSI array logic. Constructions of array logic for use in decoding are described. These codes lend themselves for use in a concatenated coding scheme.

On the number of polynomials over GF(2) that factor into 2, 3 or 4 prime polynomials

In this paper a simple method is presented to derive formulas for the number of polynomials over GF(2) which factor into two, three, and four prime polynomials only. A table is given, summarizing the above numbers for polynomials of degree up to 127. Furthermore, the computed values are compared with an asymptotic approximation for these values.

Generalizations of the spectral theorem for matrices. II. matrix polynomials over arbitrary fields

The spectral theorem for matrices is generalized to matrix polynomials over arbitrary fields. An associated theory of generalized interpolation with respect to a set of prime polynomials is developed, and Drazin inverses are used to identify the spectral components. These results are then used to state the spectral theorem for functions of real matrices.

Fault signature effectiveness of microcomputer address bus

The letter presents the analysis of the effectiveness of compact testing of a microcomputer address bus with the assumed fault model SA1 and SA0, assuming that the testing schemes use a linear feedback shift register LFSR with a prime characteristic polynomial p(x) of degree n≥2.

Diophantine equations over function fields

This paper proves the boundedness of the degrees of at least two components of an arbitrary solution of the equation , where are pairwise relatively prime polynomials. Bibliography: 4 titles.

DIOPHANTINE EQUATIONS OVER FUNCTION FIELDS

This paper proves the boundedness of the degrees of at least two components of an arbitrary solution of the equation , where are pairwise relatively prime polynomials. Bibliography: 4 titles.

High-gain decentralised control of interconnected dynamical systems

It is shown that the class of controllable and observable interconnected dynamical systems that are amenable to high-gain decentralised control can be characterised in terms of the relatively prime polynomial matrix factors of the transfer-function matrices of the disconnected subsystems. Moreover, this characterisation greatly facilitates the synthesis of high-gain decentralised controllers for such amenable systems.

Design of linear multivariable discrete-time tracking systems incorporating fast-sampling error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which ire amenable to fast-sampling error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of fast-sampling error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.

Design of linear multivariable discrete-time tracking systems incorporating fast-sampling decentralized error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which are amenable to fast-sampling decentralized error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of fast-sampling decentralized error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant. with two inputs and two outputs.

Design of linear multivariable continuous-time tracking systems incorporating high-gain decentralized error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which are amenable to high-gain decentralized error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of high-gain decentralized error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.

Some relations satisfied by prime polynomial matrices and their role in linear multivariable system theory

A number of relations which are satisfied by prime polynomial matrices are derived and then used to study the polynomial matrix equation BG_{1} + G_{2}A = V and to parametrically characterize the class of stabilizing output feedback compensators.