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A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $beta$ is one of those roots, then within one unit of $beta$ lies a root of the polynomials derivative. If we define $r(beta)$ to be the greatest possible distance between $beta$ and the closest root of the derivative, then Sendovs conjecture claims that $r(beta) le 1$. In this paper, we assume (without loss of generality) that $0 le beta le 1$ and make the stronger conjecture that $r(beta) le 1-(3/10)beta(1-beta)$. We prove this new conjecture for all polynomials of degree 2 or 3, for all real polynomials of degree 4, and for all polynomials of any degree as long as all their roots lie on a line or $beta$ is sufficiently close to 1.
We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newtons Iteration. Our method exhibits quadratic convergence when refining isolati
Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisel
In this paper, we prove a conjecture posed by Li-Yang in cite{ly3}. We prove the following result: Let $f(z)$ be a nonconstant entire function, and let $a(z) otequivinfty, b(z) otequivinfty$ be two distinct small meromorphic functions of $f(z)$. If $
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the Euler characteristic integral of a certain cohomotopy class over its scheme of fixed point
The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,bar{z})$ on $mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,bar{z}) |z|^2$ under the hypot