Related papers: Multiple discriminants and critical values of a mu…
Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen independently according to the probability measure $\mu$ on the complex plane. For a deterministic point $\xi$ lying outside the support of $\mu$, we show that almost…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear…
It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…
Assuming the existence of a sequence of exceptional discriminants of quadratic fields, we show that a hundred percent of zeros of the Riemann zeta function are on the critical line in specific segments. This is a special case of a more…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
Given a divisorial discrete valuation 'centered at infinity' on C[x,y], we show that its sign on C[x,y] (i.e. whether it is negative or non-positive on non-constant polynomials) is completely determined by the sign of its value on the 'last…
We investigate the existence and multiplicity of solutions for higher order discrete boundary value problems via critical point theory.
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the…
We consider the joint value distribution of Dirichlet $L$-functions in the critical strip $\frac{1}{2} < \sigma < 1$. We show that the values of distinct Dirichlet $L$-functions are dependent in the sense that they do not behave like…
The product of two complex-valued harmonic function is not in general complex-valued harmonic function. In this paper we show that if a complex-valued harmonic function is the product of two complex-valued harmonic functions, then it is the…
We derive a large deviation principle for the empirical measure of zeros of random polynomials with i.i.d. exponential coefficients.
We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct…
In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is…
The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that…
We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's…
We study the number of real rational degree n functions (considered up to linear fractional transformations of the independent variable) with a given set of 2n-2 distinct real critical values. We present a combinatorial reformulation of…
Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.