Related papers: An explicit univariate and radical parametrization…
The problem of determining an explicit one-parameter power form representation of the proper $n$-th degree Zolotarev polynomials on $[-1,1]$ can be traced back to P. L. Chebyshev. It turned out to be complicated, even for small values of…
We consider the classical problem of estimating norms of higher order derivatives of algebraic polynomial via the norms of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A.…
While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to…
Let $P(k,n)$ be the set of products of $k$ factors from the set $\{1,\ldots , n\}.$ In 1955, Erd\H{o}s posed the problem of determining the order of magnitude of $|P (2, n)|$ and proved that $|P (2, n)| = o(n^2 )$ for $n \to\infty$. In…
A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals…
We consider the classical problem of estimating norm of the derivative of algebraic polynomial via the norm of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by V. Markov. In this…
The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration…
In an influential 1877 paper, Zolotarev asked and answered four questions about polynomial and rational approximation. We ask and answer two questions: what are the best rational approximants $r$ and $s$ to $\sqrt{z}$ and $\mbox{sign}(z)$…
The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results of Sylvester (1851),…
We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all…
The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the…
We will show that the roots of a polynomial equation in one variable of degree n are related to the solutions of a symmetric quadratic form in n-1 variables with constant positive integer coefficients. The classic polynomial notation will…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s} \lesssim…
The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…
We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new properties of polynomials and their roots.…
The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients…
Historically in math literature there are instances where solutions have been arrived at by different authors for equation 1 for six powers on both side of equation, for different degree 2,3,4,5,6,7,8,9. See reference number 1, by A.…
A polynomial $p\in \mathbb{C}[z]$ with three finite values is called the Zolotarev polynomial. For a class of such polynomials with the given degree, given passport and simple critical points we define a \emph{combinatorial moduli space}. A…
A startlingly simple characterization of the p-norms has recently been found by Aubrun and Nechita (arXiv:1102.2618) and by Fernandez-Gonzalez, Palazuelos and Perez-Garcia. We deduce a simple characterization of the power means of order…