Related papers: Zeros of 2 by 2 Matrix Polynomials

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical…

Let $R$ be an associative ring with identity $1$. We describe all matrices in $T_n(R)$ the ring of $n\times n$ upper triangular matrices over $R$ ($n\in \mathbb{N}$), and $T_{\infty}(R)$ the ring of infinite upper triangular matrices over…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

First, we consider the equation $ax^2 - by^2 + c = 0$, with $a,b \in N*$ and $c \in Z*$, which is a generalization of Pell's equation. Here, we show that: if this equation has an integer solution and $ab$ is not a perfect square, then it…

An $n$-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most $n.$ For an arbitrary $n$-independent node set $\mathcal X$ we are interested…

We give a bound for the number of real solutions to systems of n polynomials in n variables, where the monomials appearing in different polynomials are distinct. This bound is smaller than the fewnomial bound if this structure of the…

We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new…

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether…

We reprove that the approximate degree of the OR function on n bits is Omega(sqrt(n)). We consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory. The…

We examine the maximal number of zeros a polynomial of degree at most n with constrained coefficients may have at 1. Our results are essentially sharp and extend earlier results of this variety. An interesting connection to certain…

We use a variety of computational tools to obtain a degree-$\binom{m + n - 2}{m - 1}$ polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive $m \times n$ matrix. The degree of this equation has…

Let $B_{n}(t)$ be the $n$th Stern polynomial, i.e., the $n$th term of the sequence defined recursively as $B_{0}(t)=0, B_{1}(t)=1$ and $B_{2n}(t)=tB_{n}(t), B_{2n+1}(t)=B_{n}(t)+B_{n-1}(t)$ for $n\in\N$. It is well know that $i$th…

In this paper we investigate bounds for the zeros of a bicomplex polynomial using matrix method. In particular, we find analogue of Gershgorin disk theorem, Cauchy Theorem, theorem of Fujiwara, Walsh and other theorems concerning to zeros…

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials…

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic…

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We…

We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently,…

In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…