Cyclotomic polynomial ≈ Cyclotomic polynomial
View article: A New Family of q-Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial
A New Family of q-Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial Open
We establish a new family of q -supercongruences modulo the fourth power of a cyclotomic polynomial, and give several related results. Our main ingredients are q -microscoping and the Chinese remainder theorem for polynomials.
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A basis theorem for the affine oriented Brauer category and its cyclotomic quotients Open
The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropr…
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Some New q-Congruences for Truncated Basic Hypergeometric Series Open
We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generali…
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Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time Open
In this article, we study the geometry of units and ideals of cyclotomic rings and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic as…
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A FAMILY OF -SUPERCONGRUENCES MODULO THE CUBE OF A CYCLOTOMIC POLYNOMIAL Open
We establish a family of q -supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Kar…
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THE HEIGHT OF A CLASS OF TERNARY CYCLOTOMIC POLYNOMIALS Open
Let A(n) denote the largest absolute value of the coefficients of n-th cyclotomic polynomial ${\Phi}_n(x)$ and let p < q < r be odd primes. In this note, we give an infinite family of cyclotomic polynomials ${\Phi}_{pqr}(x)$ with A(pqr) = …
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A naive approach to genuine $G$-spectra and cyclotomic spectra Open
For any compact Lie group $G$, we give a description of genuine $G$-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogous description of cyclotomic spectra in terms of na…
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On a uniqueness property of cuspidal unipotent representations Open
The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible f…
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Normal high order elements in finite field extensions based on the cyclotomic polynomials Open
We consider elements which are both of high multiplicative order and normal in extensions Fqm of the field Fq. If the extension is defined by a cyclotomic polynomial, we construct such elements explicitly and give explicit lower bounds on …
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INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS Open
We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet seri…
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CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY Open
Given any positive integer n, let A ( n ) denote the height of the n th cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A ( n ) is unbounded. We conjecture that every natural number can arise…
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Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form Open
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on ea…
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Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series Open
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-anal…
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Enumeration of Cyclic Codes Over GF(23) Open
In this paper, we investigate the number of irreducible polynomials of \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) over GF(23). First, We factorize \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) int…
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Upper bounds for cyclotomic numbers Open
Let q be a power of a prime p, let k be a nontrivial divisor of q−1 and write e=(q−1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field Fq. A general result of our study is that (a,b)≤3 for all a,b∈Z if…
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Irreducible polynomials in Int(ℤ) Open
In order to fully understand the factorization behavior of the ring Int(ℤ) = { f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to …
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$q$-Supercongruences from Jackson's $_8ϕ_7$ summation and Watson's $_8ϕ_7$ transformation Open
$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms …
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Three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial Open
In this paper, three parametric q -supercongruences for truncated very-well-poised basic hypergeometric series are proved, one of them modulo the square, the other two modulo the cube of a cyclotomic polynomial. The main ingredients of pro…
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A -SUPERCONGRUENCE MODULO THE THIRD POWER OF A CYCLOTOMIC POLYNOMIAL Open
We derive a q -supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q -microscope for supercongruences’, Adv. Math. 346 (2019), 329–358] and the q -Dixon f…
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Cyclotomic factors of necklace polynomials Open
We observe that the necklace polynomials $M_d(x) = \frac {1}{d}\sum _{e\mid d}\mu (e)x^{d/e}$ are highly reducible over $\mathbb {Q}$ with many cyclotomic factors. Furthermore, the sequence $\Phi _d(x) - 1$ of shifted cyclotomic polynomial…
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Block Structure of Cyclotomic Polynomials Open
In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them, us…
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Difference Sets from Unions of Cyclotomic Classes of Orders 12, 20, and 24 Open
Let 𝒒𝒒 be a prime of the form 𝒒𝒒 = 𝒏𝒏𝒏𝒏 + 𝟏𝟏 for integers 𝒏𝒏 ≥ 𝟏𝟏 and 𝑵𝑵 > 𝟏𝟏. For 𝒒𝒒 < 𝟏𝟏𝟏𝟏𝟓𝟓, we show that difference sets in the additive group of the field 𝐆𝐆𝑭𝑭(𝒒𝒒) are obtained from unions of cyclotomic classes of orders 𝑵𝑵 = 𝟏𝟏𝟏𝟏, 𝟐𝟐…
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Remarks on a family of complex polynomials Open
Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1,?, mn,?, and on a sequence…
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Computation of jacobi sums and cyclotomic numbers with reduced complexity Open
Jacobi sums and cyclotomic numbers are the important objects in number\ntheory. The determination of all the Jacobi sums and cyclotomic numbers of\norder $e$ are merely intricate to compute. This paper presents the lesser\nnumbers of Jacob…
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Cyclic codes from two-prime generalized cyclotomic sequences of order 6 Open
Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\ma…
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Unitary Cyclotomic Polynomials Open
See the abstract in the attached pdf.
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Coefficients of (inverse) unitary cyclotomic polynomials Open
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime …
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Mean Values of Derivatives of $L$-functions for Cyclotomic Characters in Function Fields Open
We compute the mean value of derivatives of $L$-functions for cyclotomic characters in function fields. We also show the non-vanishing property of derivatives of $L$-functions for cyclotomic characters in function fields.
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Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials Open
The maximum gap $g(f)$ of a polynomial $f$ is the maximum of the differences (gaps) between two consecutive exponents that appear in $f$. Let $Φ_{n}$ and $Ψ_{n}$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial, respec…
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Fast polynomial arithmetic in homomorphic encryption with cyclo-multiquadratic fields Open
We discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, and show how these limitations can be overcome by replacing cyclotomic fields by a family that we refer to as …