Covering by Planks and Avoiding Zeros of Polynomials
نویسندگان
چکیده
Abstract We note that the recent polynomial proofs of spherical and complex plank covering problems by Zhao Ortega-Moreno give some general information on zeros real polynomials restricted to unit sphere. As a corollary these results, we establish several generalizations celebrated Bang theorem. prove tight analog theorem for Euclidean ball an even stronger version projective space. Specifically, ball, show every nonzero $d$-variate $P$ degree $n$, there exists point in $d$-dimensional at distance least $1/n$ from zero set $P$. Using approach, also strengthening Fejes Tóth zone conjecture sphere segments, closed parts between two parallel hyperplanes. In particular, sum angular widths segments whole is $\pi $.
منابع مشابه
Some compact generalization of inequalities for polynomials with prescribed zeros
Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$, $k^2 leq rRleq R^2$ and for $Rleq r leq k$. Our results refine and generalize certain well-known polynomial inequalities.
متن کاملZeros of Interval Polynomials
Polynomial equations with perturbed coefficients arise in several areas of engineering sciences, for instance, in automatic control theory, dynamical systems, optimization and in control theory. For such equations, it is necessary to study their roots and to establish a priori estimates to define regions containing such roots. Attending to the fact that these equations can be seen as algebraic ...
متن کاملZeros of Ramanujan Polynomials
In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients originating from Ramanujan’s work. We begin by finding their number of real zeros, establishing a bound on their sizes, and determining their limiting values. Next, we prove that all nonreal zeros of Ramanujan polynomials lie on the unit circle, and are asymptotically...
متن کاملZeros of the derivatives of Faber polynomials associated with a universal covering map
For a compact set E ⊂ C containing more than two points, we study asymptotic behavior of normalized zero counting measures {μk} of the derivatives of Faber polynomials associated with E. For example if E has empty interior, we prove that {μk} converges in the weakstar topology to a measure whose support is the boundary of g(D), where g : {|z| > r} ∪ {∞} → C\E is a universal covering map such th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2022
ISSN: ['1687-0247', '1073-7928']
DOI: https://doi.org/10.1093/imrn/rnac259