Related papers: A short proof of d'Alemberts theorem

We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {\it ascending} filtration (with…

For a class of ordinary differential operators $P$ with polynomial coefficients, we give a necessary and sufficient condition for $P$ to be globally regular in $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply $u\in \cS(\R)$ (this…

Let $\{b(n):n\in\N\}$ be the sequence of coefficients in the Taylor expansion of a rational function $R(X)\in\Q(X)$ and suppose that b(n) is a perfect $d^{\rm th}$ power for all large n. A conjecture of Pisot states that one can choose a…

Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this…

Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that…

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) is an element x in R, for which there exist polynomials q(t) and s(t) such that…

The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm…

D'Alembert made the first serious attempt to prove the Fundamental Theorem of Algebra (FTA) in 1746. An elementary proof of (FTA) based on the same idea is given in Proofs from THE BOOK. We give a shorter and more transperant version of…

Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…

The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The…

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…

It is shown that the polynomial $\lambda(t) = {\rm Tr}[(A + tB)^p]$ has nonnegative coefficients when $p \leq 7$ and A and B are any two complex positive semidefinite $n \times n$ matrices with arbitrary $n$. This proofs a general…

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…

We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…

We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that…

The Casas-Alvero conjecture says that a degree $n$ complex univariate polynomial sharing a root with each of its derivative must have only one root. In this article we give three results. The first one, is that the number of possible…

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been…

In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\Q$ if and only if all of its irrational roots are real and simple. This provides an answer to a…

The field of $p$-adic numbers $\mathbb{Q}_p$ and the ring of $p$-adic integers $\mathbb{Z}_p$ are essential constructions of modern number theory. Hensel's lemma, described by Gouv\^ea as the "most important algebraic property of the…

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We consider polynomials $p(a,a^*)$ for elements $a$ with small self-commutator norm $\|[a,a^*]\| \le \delta$ and show that many properties of…