Displaying similar documents to “Reduction and specialization of polynomials”

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

Similarity:

It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer N r such that the polynomial f ( X ) / N r represents at least r distinct primes.

A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Similarity:

Let f ( x ) = x + k n - 1 x n - 1 + k n - 2 x n - 2 + + k x + k [ x ] , where 3 k n - 1 k n - 2 k k 2 k n - 1 - 3 . We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of 2 k n - 1 - 3 on the coefficients of f(x) is the best possible in this situation.

Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang (2014)

Acta Arithmetica

Similarity:

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let f ( x ) , g ( x ) p [ x ] be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies p 1 / E ( 1 + κ / ( 1 - κ ) > M > p ε , where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then | f ( [ 0 , M ] ) g ( [ 0 , M ] ) | M 1 - ε .

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

Similarity:

We consider the space of ultradifferentiable functions with compact supports and the space of polynomials on . A description of the space ( ) of polynomial ultradistributions as a locally convex direct sum is given.

Nonreciprocal algebraic numbers of small Mahler's measure

Artūras Dubickas, Jonas Jankauskas (2013)

Acta Arithmetica

Similarity:

We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials 1 + x r + + x r , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including 2 r j < r j + 1 for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct...

The factorization of f ( x ) x n + g ( x ) with f ( x ) monic and of degree 2 .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

In this paper we investigate the factorization of the polynomials f ( x ) x n + g ( x ) [ x ] in the special case where f ( x ) is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that f ( x ) is monic and linear.

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

Similarity:

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations h ( x ) = = h r ( x ) = 0 and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then f | V extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial h ( x ) = i = 1 r h ² i ( x ) σ i ( x ) , where σ i are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

Similarity:

We present some results concerning the unirationality of the algebraic variety f given by the equation N K / k ( X + α X + α ² X ) = f ( t ) , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and f contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then f is k-unirational. A similar result is proved for a broad family of quintic polynomials...

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&amp;#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

Similarity:

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that | f | C | f | ϱ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than 1 - R ( n , d ) - 1 with R ( n , d ) = d ( 3 d - 3 ) n - 1 .

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

Similarity:

Let P k be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = k a k P k . A systematic use of the basic properties (including some nonstandard ones) of the polynomials P k results in obtaining a low order of the recurrence.

Approximation by weighted polynomials in k

Maritza M. Branker (2005)

Annales Polonici Mathematici

Similarity:

We apply pluripotential theory to establish results in k concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections Σ k a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

A characterization of Eisenstein polynomials generating extensions of degree p 2 and cyclic of degree p 3 over an unramified 𝔭 -adic field

Maurizio Monge (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let p 2 be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree p 2 over p , and extend it to when the base fields K is an unramified extension of p . When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We...

Preperiodic dynatomic curves for z z d + c

Yan Gao (2016)

Fundamenta Mathematicae

Similarity:

The preperiodic dynatomic curve n , p is the closure in ℂ² of the set of (c,z) such that z is a preperiodic point of the polynomial z z d + c with preperiod n and period p (n,p ≥ 1). We prove that each n , p has exactly d-1 irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of n , p . We also compute the genus of each component and the Galois group of the defining polynomial of n , p .

Every compact set in 𝐂 n is a good compact set

Jan Erik Björk (1970)

Annales de l'institut Fourier

Similarity:

Let K be an compact subset of an open set V in C n . We show the existence of an open neighborhood U of K satisfying the following condition : if f is holomorphic in V and if there exists a sequence of polynomials which approximate f uniformly in some open neighborhood U f of K , there exists a sequence of polynomial which approximate f uniformly in U .

Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, Olivier Ramaré (2016)

Acta Arithmetica

Similarity:

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ( [ n α ] β ) n 0 , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper....

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

Similarity:

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let A m be the mth coefficient of the square f(x)² of...

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

Similarity:

Let q [ t ] be the polynomial ring over the finite field q , and let N be the subset of q [ t ] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A N for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D ( N ) q N ( l o g N ) 7 / N .

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

Similarity:

Let K be any subset of N . We define a pluricomplex Green’s function V K , θ for θ-incomplete polynomials. We establish properties of V K , θ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of N , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on K s u p p ( d d c V K , θ ) N . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute s u p p ( d d c V K , θ ) N when K is a compact...

Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials u , . . . , u m - 1 K [ X , . . . , X m - 1 ] such that f ( j = 0 m - 1 ξ j X j ) = j = 0 m - 1 ξ j u j . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then u , . . . , u m - 1 have no common divisor in K [ X , . . . , X m - 1 ] of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several...

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let · be the uniform norm in the unit disk. We study the quantities M n ( α ) : = inf ( z P ( z ) + α - α ) where the infimum is taken over all polynomials P of degree n - 1 with P ( z ) = 1 and α > 0 . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that inf α > 0 M n ( α ) = 1 / n . We find the exact values of M n ( α ) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

On the lattice of polynomials with integer coefficients: the covering radius in L p ( 0 , 1 )

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

Similarity:

The paper deals with the approximation by polynomials with integer coefficients in L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. Let P n , r be the space of polynomials of degree ≤ n which are divisible by the polynomial x r ( 1 - x ) r , r ≥ 0, and let P n , r P n , r be the set of polynomials with integer coefficients. Let μ ( P n , r ; L p ) be the maximal distance of elements of P n , r from P n , r in L p ( 0 , 1 ) . We give rather precise quantitative estimates of μ ( P n , r ; L ) for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of μ ( P n , r ; L p ) for p ≠ 2. It follows that μ ( P n , r ; L p ) n - 2 r - 2 / p as n → ∞. The results...