Results 11 to 20 of about 485 (68)
CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY
Mathematika, Volume 67, Issue 1, Page 214-234, January 2021., 2021 Abstract Given any positive integer n, let A(n) denote the height of the nth 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 as value of A(n) and prove this assuming that for every pair of consecutive primes p and p′ with p⩾127, we have ...
Alexandre Kosyak +3 more
wiley +1 more source
Multivariable polynomial injections on rational numbers [PDF]
, 2010 For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x,y) in k[x,y] inducing an injection k x k --> k.Comment: 4 ...
Poonen, Bjorn
core +3 more sources
Root separation for reducible monic polynomials of odd degree [PDF]
, 2017 We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let e_r*(d)=limsup_{deg(P)=d, h(P)->
Dujella, Andrej, Pejkovic, Tomislav
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A coprimality condition on consecutive values of polynomials [PDF]
, 2017 Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can find infinitely many integers $n\geq 0$ with the property that none of $f(n+1),f(n+2),\dots,f(n+k)$
Sanna, Carlo, Szikszai, Márton
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A construction of integer-valued polynomials with prescribed sets of lengths of factorizations [PDF]
, 2013 For an arbitrary finite set S of natural numbers greater 1, we construct an integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of lengths of f is the set of all natural numbers n, such that f has a factorization as a product of n ...
Ch Frei +4 more
core +2 more sources
Polynomials with a sharp Cauchy bound and their zeros of maximal modulus
, 2015 The moduli of zeros of a complex polynomial are bounded by the positive zero of an associated auxiliary polynomial. The bound is due to Cauchy. This note describes polynomials with a sharp Cauchy bound and the location of peripheral zeros.
H. K. Wimmer
semanticscholar +1 more source
, 2010 The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients.
Warnaar, S. Ole, Zudilin, Wadim
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A determinant formula for sums of powers of integers
, 2014 In this note, we first derive a recursive formula for the sum of powers Sk(n) = 1 k + 2k + · · · + nk, with k and n non-negative integers. We then apply it to establish, via Cramer’s rule, an explicit determinant formula for Sk(n) involving the Bernoulli
J. Cereceda
semanticscholar +1 more source
Root multiplicities and number of nonzero coefficients of a polynomial
, 2005 It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots.
Brindza B., Mattarei S., SANDRO MATTAREI
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On some special Legendre sums of the form
Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2017 We prove that sums of the form with f(X), g(X) ∈ ℤ[X] can be explicitly computed whenever f and g are subject to some certain conditions which are defined in the paper.
Andrica Dorin, Crişan Vlad
doaj +1 more source