• Title/Summary/Keyword: minimal polynomials

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MINIMAL POLYNOMIAL DYNAMICS ON THE p-ADIC INTEGERS

  • Sangtae Jeong
    • Journal of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.1-32
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    • 2023
  • In this paper, we present a method of characterizing minimal polynomials on the ring 𝐙p of p-adic integers in terms of their coefficients for an arbitrary prime p. We first revisit and provide alternative proofs of the known minimality criteria given by Larin [11] for p = 2 and Durand and Paccaut [6] for p = 3, and then we show that for any prime p ≥ 5, the proposed method enables us to classify all possible minimal polynomials on 𝐙p in terms of their coefficients, provided that two prescribed prerequisites for minimality are satisfied.

REPRESENTATION OF SOME BINOMIAL COEFFICIENTS BY POLYNOMIALS

  • Kim, Seon-Hong
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.677-682
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    • 2007
  • The unique positive zero of $F_m(z):=z^{2m}-z^{m+1}-z^{m-1}-1$ leads to analogues of $2(\array{2n\\k}\)$(k even) by using hypergeometric functions. The minimal polynomials of these analogues are related to Chebyshev polynomials, and the minimal polynomial of an analogue of $2(\array{2n\\k}\)$(k even>2) can be computed by using an analogue of $2(\array{2n\\k}\)$. In this paper we show that the analogue of $2(\array{2n\\2}\)$. In this paper we show that the analygue $2(\array{2n\\2}\)$ is the only real zero of its minimal polynomial, and has a different representation, by using a polynomial of smaller degree than $F_m$(z).

ON SELF-RECIPROCAL POLYNOMIALS AT A POINT ON THE UNIT CIRCLE

  • Kim, Seon-Hong
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1153-1158
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    • 2009
  • Given two integral self-reciprocal polynomials having the same modulus at a point $z_0$ on the unit circle, we show that the minimal polynomial of $z_0$ is also self-reciprocal and it divides an explicit integral self-reciprocal polynomial. Moreover, for any two integral self-reciprocal polynomials, we give a sufficient condition for the existence of a point $z_0$ on the unit circle such that the two polynomials have the same modulus at $z_0$.

MINIMAL QUADRATIC RESIDUE CYCLIC CODES OF LENGTH $2^{n}$

  • BATRA SUDHIR;ARORA S. K.
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.25-43
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    • 2005
  • Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

Analysis of One-dimensional cellular automata over GF(q)

  • Cho, Sung-Jin;Kim, Han-Doo;Choi, Un-Sook
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.2
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    • pp.21-32
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    • 2000
  • We study theoretical aspects of one-dimensional cellular automata over GF(q), where q is a power of a prime. Some results about the characteristic polynomials of such cellular automata are given. Intermediate boundary cellular automata are defined and related to the more common null boundary cellular automata.

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EFFICIENT ALGORITHMS FOR COMPUTING THE MINIMAL POLYNOMIALS AND THE INVERSES OF LEVEL-k Π-CIRCULANT MATRICES

  • Jiang, Zhaolin;Liu, Sanyang
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.3
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    • pp.425-435
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    • 2003
  • In this paper, a new kind of matrices, i.e., $level-{\kappa}$ II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.

CONSTRUCTION OF A SYMMETRIC SUBDIVISION SCHEME REPRODUCING POLYNOMIALS

  • Ko, Kwan Pyo
    • Communications of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.395-414
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    • 2016
  • In this work, we study on subdivision schemes reproducing polynomials and build a symmetric subdivision scheme reproducing polynomials of a certain predetermined degree, which is a slight variant of the family of Deslauries-Dubic interpolatory ones. Related to polynomial reproduction, a necessary and sufficient condition for a subdivision scheme to reproduce polynomials of degree L was recently established under the assumption of non-singularity of subdivision schemes. In case of stepwise polynomial reproduction, we give a characterization for a subdivision scheme to reproduce stepwise all polynomials of degree ${\leq}L$ without the assumption of non-singularity. This characterization shows that we can investigate the polynomial reproduction property only by checking the odd and even masks of the subdivision scheme. The minimal-support condition being relaxed, we present explicitly a general formula for the mask of (2n + 4)-point symmetric subdivision scheme with two parameters that reproduces all polynomials of degree ${\leq}2n+1$. The uniqueness of such a symmetric subdivision scheme is proved, provided the two parameters are given arbitrarily. By varying the values of the parameters, this scheme is shown to become various other well known subdivision schemes, ranging from interpolatory to approximating.

Pairing-Friendly Curves with Minimal Security Loss by Cheon's Algorithm

  • Park, Cheol-Min;Lee, Hyang-Sook
    • ETRI Journal
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    • v.33 no.4
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    • pp.656-659
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    • 2011
  • In ICISC 2007, Comuta and others showed that among the methods for constructing pairing-friendly curves, those using cyclotomic polynomials, that is, the Brezing-Weng method and the Freeman-Scott-Teske method, are affected by Cheon's algorithm. This paper proposes a method for searching parameters of pairing-friendly elliptic curves that induces minimal security loss by Cheon's algorithm. We also provide a sample set of parameters of BN-curves, FST-curves, and KSS-curves for pairing-based cryptography.

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

  • Drungilas, Paulius
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.835-840
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    • 2008
  • The main result of this paper shows that every reciprocal Littlewood polynomial, one with {-1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Little-wood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $$A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}$$ for positive integer $N{\geqslant}2$.