On the distribution of a nonnegative difference between two $χ^2$-distributed second degree polynomial statistics

J. K. Baksalary; R. Kala

Displaying similar documents to “On the distribution of a nonnegative difference between two $χ^{2}$ -distributed second degree polynomial statistics”

An alternative polynomial Daugavet property

Elisa R. Santos (2014)

Studia Mathematica

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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $m a x_{ω \in} | | I d + ω P | | = 1 + | | P | |$ . We study the stability of the APDP by c₀-, $ℓ_{\infty}$ - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty} (μ, X)$ and C(K,X), where X has the APDP.

Sum of squares and the Łojasiewicz exponent at infinity

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

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ 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 $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

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

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

Annales Polonici Mathematici

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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 ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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

Polynomial flows on $R^{n}$

Gary Hosler-Meisters (1989)

Banach Center Publications

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Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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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...

Whitney type inequality, pointwise version

Yu. A. Brudnyi, I. E. Gopengauz (2013)

Studia Mathematica

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The main result of the paper estimates the asymptotic behavior of local polynomial approximation for $L_{p}$ functions at a point via the behavior of μ-differences, a generalization of the kth difference. The result is applied to prove several new and extend classical results on pointwise differentiability of $L_{p}$ functions including Marcinkiewicz-Zygmund’s and M. Weiss’ theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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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 $A \subseteq_{N}$ 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 $D (N) ≪ q^{N} {(l o g N)}^{7} / N$ .

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ 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.

Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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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 $u ₀, . . ., u_{m - 1} \in K [X ₀, . . ., X_{m - 1}]$ such that $f (\sum_{j = 0}^{m - 1} ξ^{j} X_{j}) = \sum_{j = 0}^{m - 1} ξ^{j} u_{j}$ . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u ₀, . . ., u_{m - 1}$ have no common divisor in $K [X ₀, . . ., X_{m - 1}]$ 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...

Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

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We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K / k} (X ₁ + α X ₂ + α ² X ₃) = f (t)$ , 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...

Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, Olivier Ramaré (2016)

Acta Arithmetica

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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, ${([n α] β)}_{n \geq 0}$ , 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....

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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We show that for a polynomial mapping $F = (f ₁, . . ., f ₘ) : ℂ^{n} \to ℂ^{m}$ the Łojasiewicz exponent $_{\infty} (F)$ of F is attained on the set $z \in ℂ^{n} : f ₁ (z) \cdot . . . \cdot f ₘ (z) = 0$ .

Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang (2014)

Acta Arithmetica

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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 $f (x), g (x) \in_{p} [x]$ 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 $| f ([0, M]) \cap g ([0, M]) | ≲ M^{1 - ε}$ .

The Jacobian Conjecture in case of "non-negative coefficients"

Ludwik M. Drużkowski (1997)

Annales Polonici Mathematici

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It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F (x ₁, . . ., x_{n}) = x - H (x) : = (x ₁ - H ₁ (x ₁, . . ., x_{n}), . . ., x_{n} - H_{n} (x ₁, . . ., x_{n}))$ , where $H_{j}$ are homogeneous polynomials of degree 3 with real coefficients (or $H_{j} = 0$ ), j = 1,...,n and H’(x) is a nilpotent matrix for each $x = (x ₁, . . ., x_{n}) \in ℝ^{n}$ . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $d e g F^{- 1} \leq {(d e g F)}^{i n d F - 1}$ , where $i n d F : = m a x i n d H^{'} (x) : x \in ℝ^{n}$ . Note that the above inequality is not true when the coefficients...

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

Approximation by weighted polynomials in $ℝ^{k}$

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in $ℝ^{k}$ 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 $Σ \subset ℝ^{k}$ 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...

On a problem of Sidon for polynomials over finite fields

Wentang Kuo, Shuntaro Yamagishi (2016)

Acta Arithmetica

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Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, $l i m_{n \to \infty} r ₙ (ω) / n^{ϵ} = 0$ . P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in $_{q} [T]$ , where $_{q}$ is a finite field of...

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . 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...

Polynomial quotients: Interpolation, value sets and Waring's problem

Zhixiong Chen, Arne Winterhof (2015)

Acta Arithmetica

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For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p, w} (u)$ are defined by $q_{p, w} (u) \equiv (u^{w} - u^{w p}) / p m o d p$ with $0 \leq q_{p, w} (u) \leq p - 1$ , u ≥ 0, which are generalizations of Fermat quotients $q_{p, p - 1} (u)$ . First, we estimate the number of elements $1 \leq u < N \leq p$ for which $f (u) \equiv q_{p, w} (u) m o d p$ for a given polynomial f(x) over the finite field $_{p}$ . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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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.

Exponent of growth of polynomial mappings of $C^{2}$ into $C^{2}$

Jacek Chadzyński, Tadeusz Krasiński (1988)

Banach Center Publications

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Displaying similar documents to “On the distribution of a nonnegative difference between two χ 2 -distributed second degree polynomial statistics”

Displaying similar documents to “On the distribution of a nonnegative difference between two $χ^{2}$ -distributed second degree polynomial statistics”