Displaying similar documents to “Sum of squares and the Łojasiewicz exponent at infinity”

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...

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 .

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 .

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

Similarity:

Let P be a polynomial with integral coefficients. Shapiro showed that if the values of P at infinitely many blocks of consecutive integers are of the form Q ( m ) , where Q is a polynomial with integral coefficients, then P ( x ) = Q ( R ( x ) ) for some polynomial R . In this paper, we show that if the values of P at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form m q where q is an integer greater than 1, then P ( x ) = ( R ( x ) ) q for some polynomial R ( x ) .

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

Similarity:

We show that for a polynomial mapping F = ( f , . . . , f ) : n m the Łojasiewicz exponent ( F ) of F is attained on the set z n : f ( z ) · . . . · f ( z ) = 0 .

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

Similarity:

For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

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.

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

Similarity:

For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p 1/p , aj ∈ ℂ , such that ( x - 1 ) k divides P(x). For n ∈ ℕ and L > 0 let κ ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L m a x 1 j n | a j | , a j , such that ( x - 1 ) k divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that c 1 ( n / L ) - 1 κ ( n , L ) c 2 ( n / L ) for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

On a generalization of the Beiter Conjecture

Bartłomiej Bzdęga (2016)

Acta Arithmetica

Similarity:

We prove that for every ε > 0 and every nonnegative integer w there exist primes p 1 , . . . , p w such that for n = p 1 . . . p w the height of the cyclotomic polynomial Φ n is at least ( 1 - ε ) c w M n , where M n = i = 1 w - 2 p i 2 w - 1 - i - 1 and c w is a constant depending only on w; furthermore l i m w c w 2 - w 0 . 71 . In our construction we can have p i > h ( p 1 . . . p i - 1 ) for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

On nonsingular polynomial maps of ℝ²

Nguyen Van Chau, Carlos Gutierrez (2006)

Annales Polonici Mathematici

Similarity:

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity ( J ) : There does not exist a sequence ( p k , q k ) ² of complex singular points of F such that the imaginary parts ( ( p k ) , ( q k ) ) tend to (0,0), the real parts ( ( p k ) , ( q k ) ) tend to ∞ and F ( ( p k ) , ( q k ) ) ) a ² . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition ( J ) and if, in addition, the restriction of F to every real level set P - 1 ( c ) is proper for values of |c| large enough.

On the irreducible factors of a polynomial over a valued field

Anuj Jakhar (2024)

Czechoslovak Mathematical Journal

Similarity:

We explicitly provide numbers d , e such that each irreducible factor of a polynomial f ( x ) with integer coefficients has a degree greater than or equal to d and f ( x ) can have at most e irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field.

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...

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.

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We study the extensibility of piecewise polynomial functions defined on closed subsets of 2 to all of 2 . The compact subsets of 2 on which every piecewise polynomial function is extensible to 2 can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

Exceptional sets in Waring's problem: two squares and s biquadrates

Lilu Zhao (2014)

Acta Arithmetica

Similarity:

Let R s ( n ) denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When s = 3 or 4, it is established that the anticipated asymptotic formula for R s ( n ) holds for all n X with at most O ( X ( 9 - 2 s ) / 8 + ε ) exceptions.

Relative exactness modulo a polynomial map and algebraic ( p , + ) -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

Similarity:

Let F = ( f 1 , ... , f q ) be a polynomial dominating map from n to  q . We study the quotient 𝒯 1 ( F ) of polynomial 1-forms that are exact along the generic fibres of F , by 1-forms of type d R + a i d f i , where R , a 1 , ... , a q are polynomials. We prove that 𝒯 1 ( F ) is always a torsion [ t 1 , ... , t q ] -module. Then we determine under which conditions on F we have 𝒯 1 ( F ) = 0 . As an application, we study the behaviour of a class of algebraic ( p , + ) -actions on n , and determine in particular when these actions are trivial.

On the distribution of the roots of polynomial z k - z k - 1 - - z - 1

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We consider the polynomial f k ( z ) = z k - z k - 1 - - z - 1 for k 2 which arises as the characteristic polynomial of the k -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of f k ( z ) which lie inside the unit disk.

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let f ( x ) be a cubic, monic and separable polynomial over a field of characteristic p 3 and let E be the elliptic curve given by y 2 = f ( x ) . In this paper we prove that the coefficient at x 1 2 p ( p - 1 ) in the p –th division polynomial of E equals the coefficient at x p - 1 in f ( x ) 1 2 ( p - 1 ) . For elliptic curves over a finite field of characteristic p , the first coefficient is zero if and only if E is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

Similarity:

Let 𝒜 be the algebra of quaternions or octonions 𝕆 . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f ( t ) 𝒜 [ t ] has a root in 𝒜 . As a consequence, the Jacobian determinant | J ( f ) | is always non-negative in 𝒜 . Moreover, using the idea of the topological degree we show that a regular polynomial g ( t ) over 𝒜 has also a root in 𝒜 . Finally, utilizing multiplication ( * ) in 𝒜 , we prove various results on the topological degree...

Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Similarity:

Let f = a x + b x q + x 2 q - 1 q [ x ] . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of q ² . This result allows us to solve a related problem: Let g n , q p [ x ] (n ≥ 0, p = c h a r q ) be the polynomial defined by the functional equation c q ( x + c ) n = g n , q ( x q - x ) . We determine all n of the form n = q α - q β - 1 , α > β ≥ 0, for which g n , q is a permutation polynomial of q ² .

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

Similarity:

In the space Λ p of polynomial p-forms in ℝⁿ we introduce some special inner product. Let H p be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that Λ p splits as the direct sum d * ( Λ p + 1 ) δ * ( Λ p - 1 ) H p , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.