On polynomials with a prescribed zero

Let us denote byΛ _{n, 1} the supremum of (max_∥z∥=1∥p′ _n (z)∥)/ (max_∥z∥=1∥p _n (z)∥) taken over all polynomialsp _n of degree at mostn having a zero on the unit circle {z ∈ C∶∥z∥=1}. We show that Λn.1=n-(π ²/16)(1/n)+O(1/n ².     

Primitive polynomials with a prescribed coefficient

The Hansen–Mullen [Math. Comput. 59 (1992) 639–643, S47–S50] conjecture on primitive polynomials is established for polynomials of degree at least nine. It postulates the existence of a primitive polynomial over any finite field with any specified coefficient arbitrarily prescribed. The theory extends to polynomials of smaller degree: work is in hand to fashion a proof in these cases too.     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

An inequality for polynomials with a prescribed zero

Let p_n(z)=∑_(k-0)~n a_kz~k be a polynomial of degree n such that |p_n(z)|≤M for |z|≤1. It is well.known that for 0≤u相似文献   

Random polynomials with prescribed Newton polytope

The Newton polytope of a polynomial is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in of a system of polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in for these distributions.

Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree in complex variables with its usual SU-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope . We then determine the asymptotics of the conditional expectation of simultaneous zeros of polynomials with Newton polytope as . When , the unit simplex, it is clear that the expected zero distributions are uniform relative to the Fubini-Study form. For a convex polytope , we show that there is an allowed region on which is asymptotically uniform as the scaling factor . However, the zeros have an exotic distribution in the complementary forbidden region and when (the case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as .

     

Counting irreducible polynomials with prescribed coefficients over a finite field

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalence classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.     

Two extremal problems for trigonometric polynomials with a prescribed number of harmonics

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 12–18, January, 1991.     

On permutation polynomials of prescribed shape

We count permutation polynomials of ${\mathbb{F}}_q$ which are sums of $m+1$ $(?2)$ monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.     

On the enumeration of polynomials with prescribed factorization pattern

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomials of degree m over a finite field, as well as the number of monic n-smooth polynomials of degree m with the prescribed trace coefficient.     

Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity

For $k\in \{1,2,3,\dots \}$, we construct an even compactly supported piecewise polynomial ${\psi }_k$ whose Fourier transform satisfies $A_k{(1+{\omega }^2)}^{?k}\le {\stackrel{?}{\psi }}_k\left(\omega \right)\le B_k{(1+{\omega }^2)}^{?k}$, $\omega \in \mathbb{R}$, for some constants $B_k\ge A_k>0$. The degree of ${\psi }_k$ is shown to be minimal, and is strictly less than that of Wendland’s function ${?}_{1,k?1}$ when $k>2$. This shows that, for $k>2$, Wendland’s piecewise polynomial ${?}_{1,k?1}$ is not of minimal degree if one places no restrictions on the number of pieces.     

The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

For a matrix polynomial P(λ) and a given complex number μ, we introduce a (spectral norm) distance from P(λ) to the matrix polynomials that have μ as an eigenvalue of geometric multiplicity at least κ, and a distance from P(λ) to the matrix polynomials that have μ as a multiple eigenvalue. Then we compute the first distance and obtain bounds for the second one, constructing associated perturbations of P(λ).     

The existence of matrices with prescribed characteristic and permanental polynomials

Let d(λ) and p(λ) be monic polynomials of degree n?2 with coefficients in F, an algebraically closed field or the field of all real numbers. Necessary and sufficient conditions for the existence of an n-square matrix A over F such that det(λI?A)=d(λ) and per(λI?A=p(λ) are given in terms of the coefficients of d(λ) and p(λ).