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We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root…

Numerical Analysis · Computer Science 2008-09-05 Deepak Ponvel Chermakani

A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real…

General Mathematics · Mathematics 2021-06-21 Arindama Singh

Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…

Algebraic Geometry · Mathematics 2013-04-23 Alicia Dickenstein , Ioannis Emiris , Anna Karasoulou

We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive…

Classical Analysis and ODEs · Mathematics 2023-01-24 Vladimir Petrov Kostov

We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…

Classical Analysis and ODEs · Mathematics 2023-07-04 Xuezhi Chen , Jingwei Guo

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…

Numerical Analysis · Mathematics 2014-07-01 Victor Y. Pan

We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Mar\'ik for a…

Complex Variables · Mathematics 2019-09-04 Mario DeFranco

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

Rings and Algebras · Mathematics 2025-01-07 Alina G. Goutor

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

Symbolic Computation · Computer Science 2017-04-14 Victor Y. Pan , Liang Zhao

An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this…

Rings and Algebras · Mathematics 2026-05-26 Jason Bland , Skip Garibaldi , Joel Rosenberg

For a real polynomial $p = \sum_{i=0}^{n} c_ix^i$ with no negative coefficients and $n\geq 6$, let $\beta (p) = \inf_{i=1}^{n-1} c_i^2/c_{i+1}c_{i-1}$ (so $\beta (p) \geq 1$ entails that $p$ is log concave). If $\beta(p) > 1.45...$, then…

Classical Analysis and ODEs · Mathematics 2010-10-01 David Handelman

Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$…

Combinatorics · Mathematics 2008-09-10 Yi Wang , Yeong-Nan Yeh

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Number Theory · Mathematics 2009-07-16 L. Bary-Soroker

In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…

Algebraic Geometry · Mathematics 2021-10-19 Marc Maliar

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.

Combinatorics · Mathematics 2019-02-04 J. -M Billiot , E Fontenas

In this paper we use Dirichlet's theorem in order to elementally prove two theorems. The first says that since a polynomial ax+b generates one prime, it also generates infinites. The second theorem (which is proved in a very simillar way to…

General Mathematics · Mathematics 2014-05-23 Hilário Fernandes

A classical theorem due to Borel asserts that any formal serie with real coefficients is the Taylor expansion of a germ of $\mathcal{C}^{\infty}- {\rm function}$. We study such a problem in the context of Lie algebras of vector fields or of…

Dynamical Systems · Mathematics 2017-10-26 D. Cerveau , D. Garba Belko

Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…

Numerical Analysis · Computer Science 2014-09-09 Bahman Kalantari

Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on…

Logic in Computer Science · Computer Science 2018-11-28 Wenda Li , Lawrence C. Paulson

The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…

General Mathematics · Mathematics 2011-10-28 Konstantine Zelator