Related papers: Root structures of polynomials with sparse exponen…
In this paper I consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected…
After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their…
A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
We study linear transformations $T \colon \mathbb{R}[x] \to \mathbb{R}[x]$ of the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is a real orthogonal polynomial system. Such transformations that preserve or shrink the location of the complex zeros…
This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this paper's main result in positive characteristic: If $F$ is a…
We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") and the…
Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral…
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…
We prove two recent conjectures of Bourn and Erickson (2023) regarding the real-rootedness of a certain family of polynomials $N_n(t)$ as well as the sum of their coefficients. These polynomials arise as the numerators of generating…
We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most $d$. For this, we interpret the basis polynomials as vector…
Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact…
The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in…
We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several…
We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$…
We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate…
This paper investigates the stratification of the discriminant hypersurface associated with a univariate polynomial via the number of its distinct complex roots. We introduce two novel approaches different from the one based on…
We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated…