Divisor Polynomial Research Articles

수학영재들이 조립제법 일반화 과정을 유추하는 사례연구

The synthetic division in school mathematics is introduced as a method to easily obtain the quotient and remainder in the division of polynomials, but it is limited to when the divisor is a first-order polynomial and is handled mechanically based only on calculation convenience. Accordingly, in this paper, we studied how analogical thinking is expressed in synthetic divison and generalization of synthetic division in long divison for mathematically giftedstudents. In mathematically gifted students when the divisor is a quadratic polynomial and further in the generalization of synthetic division. At first, mathematically gifted students were unable to find the quotient and remainder using long division method when the divisor polynomial was quadratic, and found the quotient and remainder using the concept of identity. However, by analyzing the principle of long division process and comparing it with synthetic division method, they were able to find the quotient and remainder when the divisor polynomial was quadratic. By analogy, synthetic division was completed. Later, in the generalization of synthetic division, the principle of synthetic division was described and a method for calculating the quotient and remainder was successfully presented. Therefore, the analogy is positive for developing mathematical thinking and fostering an investigative attitude to understand the detailed principles of synthetic division by analyzing long division process, breaking away from the simple mechanical synthetic division when the divisor is a linear polynomial and developing an attitude of inquiry that can make generalizations.

EM− Graded Rings

The main goal of this article is to introduce the concept of EM − G− graded rings. This concept is an extension of the notion of EM− rings. Let G be a group, and R be a G− graded commutative ring. The G− gradation of R can be extended to R[x] by taking the components (R[x])σ = Rσ[x]. We define R to be an EM − G− graded ring if every homogeneous zero divisor polynomial has an annihilating content. We provide examples of EM − G− graded rings that are not EM− rings, and we prove some interesting results regarding these rings.

Efficient BCH Code Encoding and Decoding Algorithm With Divisor-Distance-Based Polynomial Division for STT-MRAM

Error-correcting codes (ECCs) are important and widely implemented in memories from high-speed SRAM cache to high-volume 3D-NAND. However, from the viewpoint, ECC technology of Spin-transfer torque magnetic random-access memory (STT-MRAM) is not established yet, as the operation speed of STT-MRAM is higher than DRAM/3D-NAND, and its volume is larger than SRAM. Moreover, it is difficult for existing ECCs to guarantee low latency and low hardware complexity while achieving high error correction capabilities. In this work, the time efficiency of both encoding and decoding of Bose-Chaudhuri-Hocquenghem (BCH) codes for STT-MRAM is optimized. A divisor-distance-based (DDB) polynomial division method is proposed to accelerate the polynomial division of BCH encoding and decoding. The DDB division method leverages the characteristics of the divisor polynomial to achieve parallel-processing for multi-bit on the word-line in the same block in a simple manner. And DDB polynomial division method can execute in multiple approaches with different hardware architectures, where the DDB divider with a parallel multiplier (PM) has the lowest hardware complexity. To demonstrate the superiority of the proposed algorithm: The time efficiency of the proposed BCH codes is successfully verified in software, proposed algorithm increases the time efficiency of encoding by more than 10 times, and the decoding increased by about 10%. The hardware implementation of proposed DDB divider is presented for improving the hardware complexity, the PM-type DDB divider architecture eliminates the large memory block of the traditional lookup-table type parallel divider.

Supersingular loci from traces of Hecke operators

A classical observation of Deligne shows that, for any prime p ≥ 5 p \geq 5 , the divisor polynomial of the Eisenstein series E p − 1 ( z ) E_{p-1}(z) mod p p is closely related to the supersingular polynomial at p p , S p ( x ) ≔ ∏ E / F ¯ p supersingular ( x − j ( E ) ) ∈ F p [ x ] . \begin{align*} S_p(x) ≔\prod _{E/\overline {\mathbb {F}}_p \text { supersingular}}(x-j(E))\,\, \in \mathbb {F}_p[x]. \end{align*} Deuring, Hasse, and Kaneko and Zagier [Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Providence, RI, 1998, pp. 97–126] found other families of modular forms which also give the supersingular polynomial at p p . In a new approach, we prove an analogue of Deligne’s result for the Hecke trace forms T k ( z ) T_k(z) defined by the Hecke action on the space of cusp forms S k S_k . We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p p , and then relate their divisor polynomials to S p ( x ) S_p(x) using Deligne’s observation.

Blind Estimation of Self-Synchronous Scrambler Using Orthogonal Complement Space in DSSS Systems

In a non-cooperative context, a receiver has to estimate the communication parameters without any prior knowledge of the transmitter, which is highly demanding. Estimating a self-synchronous scrambler is even more challenging because the scrambling sequence of the self-synchronous scrambler is affected by the input sequence. This paper proposes an improved algorithm for blind estimation of a self-synchronous scrambler using the orthogonal property without any bias condition of the received signals in direct sequence spread spectrum (DSSS) systems. We first examine the linear relation of a scrambling sequence using the repetitive property of the spreading code used in a DSSS system. Using the obtained linear relation and the basis of the orthogonal complement space, we then acquire the feedback polynomial candidates of the scrambler. Finally, by calculating the greatest common divisor polynomial of the feedback polynomial candidates, we estimate the correct feedback polynomial. Through computer simulations, we verify that the proposed method achieves superior estimation performance compared to the existing method. Furthermore, we show that the proposed method has practically acceptable computational complexity. For these reasons, it is expected that the proposed method can be applied to blind estimation of a self-synchronous scrambler in a practical non-cooperative system.

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of supersingular elliptic curves over {overline{{mathbb {F}}_{p}}}. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form v_E. This allows us to prove that if the root number of E is -1 then all supersingular j-invariants of elliptic curves defined over {mathbb {F}}_{p} are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in {mathbb {F}}_p seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of v_E corresponding to supersingular elliptic curves defined over {mathbb {F}}_p are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.

Okutsu invariants and Newton polygons

Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in terms of certain Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to F(x); these are monic irreducible polynomials sufficiently close to F(x) to share all its Okutsu invariants. This perspective widens the scope of applications of Montes' algorithm, which can be reinterpreted as a tool to compute the Okutsu polynomials and a Montes approximation, for each irreducible factor of a monic separable polynomial f(x) in O[x].

A Finite Field Framework for Modeling, Analysis and Control of Finite State Automata

In this paper, we address the modeling, analysis and control of finite state automata, which represent a standard class of discrete event systems. As opposed to graph theoretical methods, we consider an algebraic framework that resides on the finite field <formula form="inline">${\\Op F}_2$</formula> which is defined on a set of two elements with the operations addition and multiplication, both carried out modulo 2. The key characteristic of the model is its functional completeness in the sense that it is capable of describing most of the finite state automata in use, including non-deterministic and partially defined automata. Starting from a graphical representation of an automaton and applying techniques from Boolean algebra, we derive the transition relation of our finite field model. For cases in which the transition relation is linear, we develop means for treating the main issues in the analysis of the cyclic behavior of automata. This involves the computation of the elementary divisor polynomials of the system dynamics, and the periods of these polynomials, which are shown to completely determine the cyclic structure of the state space of the underlying linear system. Dealing with non-autonomous linear systems with inputs, we use the notion of feedback in order to specify a desired cyclic behavior of the automaton in the closed loop. The computation of an appropriate state feedback is achieved by introducing an image domain and adopting the well-established polynomial matrix method to linear discrete systems over the finite field <formula form="inline">${\\Op F}_2$</formula>. Examples illustrate the main steps of our method.

Modeling and analyzing finite state automata in the finite field F2

A method for determining multilinear state space models for general finite state automata is presented. The obtained model resides on F 2 , the finite field of characteristic 2 with the operations addition and multiplication, both carried out modulo 2. It is functionally complete in the sense that it is capable of describing all finite state automata, including non-deterministic and partially defined automata. For those cases in which the model over F 2 is linear, means for a complete analysis of the cyclic behavior of these automata are recalled. With respect to these linear models, the cyclic structure of the state space is shown to be determined only by the periods of the elementary divisor polynomials of the system dynamics. An example illustrates the analysis procedure.

Gauss–Kronrod quadrature formulae for weight functions of Bernstein–Szegö type

We study the Kronrod extensions of Gaussian quadrature rules whose weight functions on [−1, 1] consist of any one of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [−1, 1]. We show that in almost all cases these extended “Gauss–Kronrod” quadrature rules have all the desirable properties: Kronrod nodes interlacing with Gauss nodes, all nodes contained in [−1, 1], and all weights positive and representable by semiexplicit formulas. Exceptions to these properties occur only for small values of n (the number of Gauss nodes), namely n ⩽ 3, and are carefully identified. The precise degree of exactness of each of these Gauss–Kronrod formulae is determined and shown to grow like 4n, rather than 3n, as is normally the case. Our findings are the result of a detailed analysis of the underlying orthogonal polynomials and “Stieltjes polynomials”. The paper concludes with a study of the limit case of a linear divisor polynomial in the weight function.

Computational algorithms for algebraic operations on multidimensional polynomials

Algebraic operations on m-D (m-dimensional) polynomials (m≥ 2) are considered. In particular the addition, multiplication and division of such polynomials are described by means of algorithms in appropriate computational form. The special case of division of a multidimensional polynomial by a polynomial of lower order in each variable is examined, and, apart from the corresponding algorithm, necessary and sufficient conditions for the existence of the quotient and a remainder polynomial of lower order than the divisor are introduced. These conditions can be formulated by inspection of the coefficients of the dividend and divisor polynomials.

An algorithm for inverting polynomial matrices

Abstract This paper gives an algorithm for computing the inverses of polynomial matrices. The Gauss-Jordan inversion method, which is commonly used for the numerical inversion of constant matrices, can be applied to the inversion of polynomial matrices. It requires, however, operations with polynomials. Moreover, in the division of the operations, if the common factors in the divisor polynomials and the dividend polynomials are not eliminated, the resultant inverse contains polynomials of high degree in the numerators and the denominators. An algorithm given in this paper requires only operations with constant matrices. This algorithm gives an inverse in the minimal degree form if a polynomial matrix to be inverted is not a special form. It is shown that this is faster than existing ones. Also, it is an extension of the Souriau-Frame-Faddeev one. Some examples are illustrated to show the feasibility of the algorithm.

Synthesis of multiple loop feedback systems Part II†

Synthesis of all-pole functions has already been demonstrated. The paper describes the extension of the method to realize various types of rational functions which are used in filter networks. The method is demonstrated on known networks and in the synthesis of a new class of parallel systems. The technique of numerator inclusion by choice of divisor polynomial and ‘ branch introduction ’ are shown. The transformation from the nodal matrix to the node matrix for the parallel systems is formulated and Number of Introductions Rule for such systems is given.

An investigation of the effect of the divisor polynomial on the responses of R-C active networks designed by the Yanagisawa procedure

An investigation of the effect of the divisor polynomial on the responses of R-C active networks designed by the Yanagisawa procedure

XXIII.—Studies in Practical Mathematics. VII. On the Theory of Methods of Factorizing Polynomials by Iterated Division

SynopsisThe division of one polynomial by another is studied with the object of ascertaining the errors produced in the coefficients of successive remainders by small errors in the coefficients of the divisor. It is shown that the matrix which effects this transformation of errors is a polynomial in the rational canonical matrix for which the divisor polynomial is characteristic. The theory gives rise to a numerous class of iterative processes for finding an exact factor, such as the extant method based on the penultimate remainder, Bairstow's iterative method of finding a quadratic factor, and many others. Some new suggestions are made for accelerating convergence.