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Journal articles on the topic 'Prime polynomials'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Beardon, Alan F. "Prime matrices and prime polynomials." Mathematical Gazette 93, no. 528 (November 2009): 433–40. http://dx.doi.org/10.1017/s0025557200185171.

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In an earlier paper in the Gazette the authors of define what it means for a matrix in a set M of n × n matrices to be prime, namely if it is not the product of two matrices in M, neither of which is the identity. They then showed that there are exactly two primes in the set M2 of 2 × 2 matrices with non-negative integral entries and unit determinant, namely,

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2

McLean, K. Robin. "Prime-Valued Polynomials." Mathematical Gazette 82, no. 494 (July 1998): 195. http://dx.doi.org/10.2307/3620402.

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3

Turnwald, Gerhard. "On Schur's conjecture." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 58, no. 3 (June 1995): 312–57. http://dx.doi.org/10.1017/s1446788700038349.

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AbstractWe study polynomials over an integral domainRwhich, for infinitely many prime idealsP, induce a permutation ofR/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field ofR. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

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4

Kao, P. H. "Almost-prime polynomials at prime arguments." Journal of Number Theory 184 (March 2018): 85–106. http://dx.doi.org/10.1016/j.jnt.2017.08.011.

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5

Bonciocat, Anca Iuliana, Nicolae Ciprian Bonciocat, and Mihai Cipu. "Irreducibility Criteria for Compositions and Multiplicative Convolutions of Polynomials with Integer Coefficients." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 1 (December 10, 2014): 73–84. http://dx.doi.org/10.2478/auom-2014-0007.

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AbstractWe provide irreducibility criteria for multiplicative convolutions of polynomials with integer coefficients, that is, for polynomials of the form hdeg f · f(g/h), where f, g, h are polynomials with integer coefficients, and g and h are relatively prime. The irreducibility conditions are expressed in terms of the prime factorization of the leading coefficient of the polynomial hdeg f · f(g/h), the degrees of f, g, h, and the absolute values of their coefficients. In particular, by letting h = 1 we obtain irreducibility conditions for compositions of polynomials with integer coefficients.

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6

Salas, Christian. "Cantor Primes as Prime-Valued Cyclotomic Polynomials." International Journal of Open Problems in Computer Science and Mathematics 5, no. 3 (September 2012): 68–74. http://dx.doi.org/10.12816/0006120.

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7

Hashemi, E. "Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–8. http://dx.doi.org/10.1155/2008/835605.

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We first study connections betweenα-compatible ideals ofRand related ideals of the skew Laurent polynomials ringR[x,x−1;α], whereαis an automorphism ofR. Also we investigate the relationship ofP(R)andNr(R)ofRwith the prime radical and the upper nil radical of the skew Laurent polynomial rings. Then by using Jordan's ring, we extend above results to the case whereαis not surjective.

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8

Akbary, Amir, and Keilan Scholten. "Artin prime producing polynomials." Mathematics of Computation 84, no. 294 (December 2, 2014): 1861–82. http://dx.doi.org/10.1090/s0025-5718-2014-02902-5.

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9

Grosswald, Emil. "On prime representing polynomials." Proceedings of the Indian Academy of Sciences - Section A 97, no. 1-3 (December 1987): 75–84. http://dx.doi.org/10.1007/bf02837816.

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10

Mikaelian, Vahagn. "On Degrees of Modular Common Divisors and the Big PrimegcdAlgorithm." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/3262450.

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We consider a few modifications of the Big prime modulargcdalgorithm for polynomials inZ[x]. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant, and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifications are used to suggest improved algorithms forgcdcalculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular on polynomials from Knuth’s example of intermediate expression swell.

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11

Fadil, Lhoussain El. "Factorization of polynomials over valued fields based on graded polynomials." Mathematica Slovaca 70, no. 4 (August 26, 2020): 807–14. http://dx.doi.org/10.1515/ms-2017-0393.

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AbstractIn this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

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12

Mollin, R. A., and H. C. Williams. "Quadratic Non-Residues and Prime-Producing Polynomials." Canadian Mathematical Bulletin 32, no. 4 (December 1, 1989): 474–78. http://dx.doi.org/10.4153/cmb-1989-068-1.

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AbstractWe will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

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13

Akishin, Aleksandr V. "Group polynomials over rings." Discrete Mathematics and Applications 30, no. 6 (December 16, 2020): 357–64. http://dx.doi.org/10.1515/dma-2020-0033.

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AbstractWe consider polynomials over rings such that the polynomials represent Latin squares and define a group operation over the ring. We introduce the notion of a group polynomial, describe a number of properties of these polynomials and the groups generated. For the case of residue rings$\mathbb{Z}_{r^n},$where r is a prime number, we give a description of groups specified by polynomials and identify a class of group polynomials that can be used to construct controlled cryptographic transformations.

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14

Chbili, Nafaa. "Graph polynomials and symmetries." Journal of Algebra and Its Applications 18, no. 09 (July 17, 2019): 1950172. http://dx.doi.org/10.1142/s021949881950172x.

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In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

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15

Zhang, Qifan. "Witt Rings and Permutation Polynomials." Algebra Colloquium 12, no. 01 (March 2005): 161–69. http://dx.doi.org/10.1142/s1005386705000155.

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Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over ℤ/p2ℤ and 3-tuples of polynomial functions over ℤ/pℤ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over ℤ/plℤ.

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16

Irving, A. J. "Almost-prime values of polynomials at prime arguments." Bulletin of the London Mathematical Society 47, no. 4 (June 12, 2015): 593–606. http://dx.doi.org/10.1112/blms/bdv035.

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17

Bary-Soroker, Lior, and Jakob Stix. "Cubic Twin Prime Polynomials are Counted by a Modular Form." Canadian Journal of Mathematics 71, no. 6 (January 9, 2019): 1323–50. http://dx.doi.org/10.4153/cjm-2018-018-9.

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AbstractWe present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

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18

Pritsker, Igor E. "The Gelfond–Schnirelman Method in Prime Number Theory." Canadian Journal of Mathematics 57, no. 5 (October 1, 2005): 1080–101. http://dx.doi.org/10.4153/cjm-2005-041-8.

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AbstractThe original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshev-type lower bound in prime number theory. We study a generalization of thismethod for polynomials inmany variables. Ourmain result is a lower bound for the integral of Chebyshev's ψ-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

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19

KIM, SE-GOO. "ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 07 (July 2009): 973–84. http://dx.doi.org/10.1142/s0218216509007300.

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Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

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20

Filaseta, Michael. "Prime values of irreducible polynomials." Acta Arithmetica 50, no. 2 (1988): 133–45. http://dx.doi.org/10.4064/aa-50-2-133-145.

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21

Murty, M. Ram. "Prime Numbers and Irreducible Polynomials." American Mathematical Monthly 109, no. 5 (May 2002): 452. http://dx.doi.org/10.2307/2695645.

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22

Lord, Nick. "79.58 Prime Values of Polynomials." Mathematical Gazette 79, no. 486 (November 1995): 572. http://dx.doi.org/10.2307/3618098.

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23

Murty, M. Ram. "Prime Numbers and Irreducible Polynomials." American Mathematical Monthly 109, no. 5 (May 2002): 452–58. http://dx.doi.org/10.1080/00029890.2002.11919872.

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24

Wu, Jie, and Ping Xi. "Quadratic polynomials at prime arguments." Mathematische Zeitschrift 285, no. 1-2 (September 3, 2016): 631–46. http://dx.doi.org/10.1007/s00209-016-1737-3.

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25

Pakovich, F. "Prime and composite Laurent polynomials." Bulletin des Sciences Mathématiques 133, no. 7 (October 2009): 693–732. http://dx.doi.org/10.1016/j.bulsci.2009.06.003.

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26

Yeh, Chi-Tsuen, and Chen-Lian Chuang. "Nil Polynomials of Prime Rings." Journal of Algebra 186, no. 3 (December 1996): 781–92. http://dx.doi.org/10.1006/jabr.1996.0394.

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27

Higa, Ryuji. "An estimation for the ascending numbers of knots by Γ-polynomials." Journal of Knot Theory and Its Ramifications 29, no. 01 (January 2020): 1950096. http://dx.doi.org/10.1142/s0218216519500962.

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For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the [Formula: see text]-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using [Formula: see text]-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using [Formula: see text]-polynomials, Conway polynomials, and the determinants.

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28

BIRMAJER, DANIEL, JUAN B. GIL, and MICHAEL WEINER. "FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ." International Journal of Number Theory 08, no. 07 (August 28, 2012): 1763–76. http://dx.doi.org/10.1142/s1793042112501011.

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We consider polynomials with integer coefficients and discuss their factorization properties in ℤ[[x]], the ring of formal power series over ℤ. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over ℤ[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

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29

CHEN, SHENSHI, YAQING CHEN, and QUANHAI YANG. "TOWARD RANDOMIZED TESTING OF q-MONOMIALS IN MULTIVARIATE POLYNOMIALS." Discrete Mathematics, Algorithms and Applications 06, no. 02 (March 19, 2014): 1450016. http://dx.doi.org/10.1142/s1793830914500165.

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Given any fixed integer q ≥ 2, a q-monomial is of the format [Formula: see text] such that 1 ≤ sj ≤ q - 1, 1 ≤ j ≤ t. q-monomials are natural generalizations of multilinear monomials. Recent research on testing multilinear monomials and q-monomials for prime q in multivariate polynomials relies on the property that Zq is a field when q ≥ 2 is prime. When q > 2 is not prime, it remains open whether the problem of testing q-monomials can be solved in some compatible complexity. In this paper, we present a randomized O*(7.15k) algorithm for testing q-monomials of degree k that are found in a multivariate polynomial that is represented by a tree-like circuit with a polynomial size, thus giving a positive, affirming answer to the above question. Our algorithm works regardless of the primality of q and improves upon the time complexity of the previously known algorithm for testing q-monomials for prime q > 7.

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30

Bonciocat, Anca Iuliana, and Nicolae Ciprian Bonciocat. "The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value." Canadian Mathematical Bulletin 52, no. 4 (December 1, 2009): 511–20. http://dx.doi.org/10.4153/cmb-2009-052-9.

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AbstractWe use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

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31

Charles, Denis, and Kristin Lauter. "Computing Modular Polynomials." LMS Journal of Computation and Mathematics 8 (2005): 195–204. http://dx.doi.org/10.1112/s1461157000000954.

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AbstractThis paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

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32

BOBER, J. W., D. FRETWELL, G. MARTIN, and T. D. WOOLEY. "SMOOTH VALUES OF POLYNOMIALS." Journal of the Australian Mathematical Society 108, no. 2 (February 1, 2019): 245–61. http://dx.doi.org/10.1017/s1446788718000320.

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Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$, there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$.

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33

Guevara Hernández, María de los Angeles, and Hugo Cabrera Ibarra. "Infinite families of prime knots with alt(K) = 1 and their Alexander polynomials." Journal of Knot Theory and Its Ramifications 28, no. 02 (February 2019): 1950010. http://dx.doi.org/10.1142/s021821651950010x.

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In this paper, we construct, by using the Alexander polynomial, infinite families of nonalternating prime knots, which have alternation number equal to one. More specifically these knots after one crossing change yield a 2-bridge knot or the trivial knot. In particular, we display two infinite families of nonalternating knots and their Alexander polynomials. Moreover, we give formulae to obtain the Conway and Alexander polynomials of oriented 3-tangles and the links formed from their closure with a specific orientation. In particular, we propose a construction to form families of links for which their Alexander polynomials can be obtained by nonrecursive formulae.

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34

Franze, C. S., and P. H. Kao. "Almost-prime values of reducible polynomials at prime arguments." Journal of Number Theory 210 (May 2020): 292–312. http://dx.doi.org/10.1016/j.jnt.2019.09.013.

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35

Polak, Jason K. C. "Counting Separable Polynomials in ℤ/n[x]." Canadian Mathematical Bulletin 61, no. 2 (June 1, 2018): 346–52. http://dx.doi.org/10.4153/cmb-2017-013-4.

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AbstractFor a commutative ring R, a polynomialf ∈ R[x] is called separable if R[x]/f is a separable R-algebra. We derive formulae for the number of separable polynomials when R = /n, extending a result of L. Carlitz. For instance, we show that the number of polynomials in /n[x] that are separable is ϕ(n)nd Πi(1 − ), where n = is the prime factorisation of n and ϕ is Euler’s totient function.

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36

BIRMAJER, DANIEL, JUAN B. GIL, and MICHAEL D. WEINER. "FACTORIZATION OF QUADRATIC POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ." Journal of Algebra and Its Applications 06, no. 06 (December 2007): 1027–37. http://dx.doi.org/10.1142/s021949880700265x.

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We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring ℤ[[x]] of formal power series over the integers. In particular, for polynomials of the form pn + pm βx + αx2 with n,m ≥ 1 and p prime, we show that reducibility in ℤ[[x]] is equivalent to reducibility in ℤp[x], the ring of polynomials over the p-adic integers.

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37

NATHANSON, MELVYN B. "CANTOR POLYNOMIALS FOR SEMIGROUP SECTORS." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350165. http://dx.doi.org/10.1142/s021949881350165x.

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A packing function on a set Ω in Rn is a one-to-one correspondence between the set of lattice points in Ω and the set N0 of non-negative integers. It is proved that if r and s are relatively prime positive integers such that r divides s - 1, then there exist two distinct quadratic packing polynomials on the sector {(x, y) ∈ R2 : 0 ≤ y ≤ rx/s}. For the rational numbers 1/s, these are the unique quadratic packing polynomials. Moreover, quadratic quasi-polynomial packing functions are constructed for all rational sectors.

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38

Günther, Christian, and Kai-Uwe Schmidt. "Lq Norms of Fekete and Related Polynomials." Canadian Journal of Mathematics 69, no. 4 (August 1, 2017): 807–25. http://dx.doi.org/10.4153/cjm-2016-023-4.

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AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

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39

KOLEY, BISWAJIT, and SATYANARAYANA REDDY ARIKATLA. "CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS." Bulletin of the Australian Mathematical Society 100, no. 1 (March 28, 2019): 41–47. http://dx.doi.org/10.1017/s0004972719000339.

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A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

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40

Chabert, Jean-Luc. "Une Caractérisation des Polynômes Prenant des Valeurs Entières Sur Tous les Nombres Premiers." Canadian Mathematical Bulletin 39, no. 4 (December 1, 1996): 402–7. http://dx.doi.org/10.4153/cmb-1996-048-7.

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AbstractWe give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.

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41

Bodin, Arnaud, Pierre Dèbes, and Salah Najib. "Prime and coprime values of polynomials." L’Enseignement Mathématique 66, no. 1 (October 6, 2020): 173–86. http://dx.doi.org/10.4171/lem/66-1/2-9.

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42

Chen, Yong-Gao, and Imre Ruzsa. "Prime values of reducible polynomials, I." Acta Arithmetica 95, no. 2 (2000): 185–93. http://dx.doi.org/10.4064/aa-95-2-185-193.

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43

Chen, Yong-Gao, Gabor Kun, Gabor Pete, Imre Z. Ruzsa, and Adam Timar. "Prime values of reducible polynomials, II." Acta Arithmetica 104, no. 2 (2002): 117–27. http://dx.doi.org/10.4064/aa104-2-2.

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44

Browkin, J., and A. Schinzel. "Prime factors of values of polynomials." Colloquium Mathematicum 122, no. 1 (2011): 135–38. http://dx.doi.org/10.4064/cm122-1-12.

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45

Benjamin, Arthur T., and Curtis D. Bennett. "The Probability of Relatively Prime Polynomials." Mathematics Magazine 80, no. 3 (June 2007): 196–202. http://dx.doi.org/10.1080/0025570x.2007.11953481.

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46

Dearden, Bruce, and Jerry Metzger. "Roots of Polynomials Modulo Prime Powers." European Journal of Combinatorics 18, no. 6 (August 1997): 601–6. http://dx.doi.org/10.1006/eujc.1996.0124.

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47

Marubayashi, Hidetoshi, Yang Lee, and Jae Keol Park. "Polynomials determining hereditary prime PI-rings." Communications in Algebra 20, no. 9 (January 1992): 2503–11. http://dx.doi.org/10.1080/00927879208824475.

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48

Alves, Sérgio M., Antônio P. Brandão, and Plamen Koshlukov. "Graded Central Polynomials forT-Prime Algebras." Communications in Algebra 37, no. 6 (June 4, 2009): 2008–20. http://dx.doi.org/10.1080/00927870802266482.

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49

McCurley, Kevin S. "Polynomials with no small prime values." Proceedings of the American Mathematical Society 97, no. 3 (March 1, 1986): 393. http://dx.doi.org/10.1090/s0002-9939-1986-0840616-4.

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50

Conrey, J. B., D. W. Farmer, and P. J. Wallace. "Factoring Hecke polynomials modulo a prime." Pacific Journal of Mathematics 196, no. 1 (November 1, 2000): 123–30. http://dx.doi.org/10.2140/pjm.2000.196.123.

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