Related papers: A remark about polynomials with specified local mi…

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative…

We prove that singular vectors have measure zero with respect to any friendly measure on $\Bbb R^n$ (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud.…

Let $(M,\omega)$ be an almost symplectic manifold ($\omega$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_X\omega=0,d(i(X)\omega)=0$, and it is Hamiltonian if, furthermore, the…

It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel…

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…

It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector fields…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

In the context of orthogonal polynomials in the plane we introduce the notion of a polynomially small (PS) perturbation of a measure. In such a case we establish relative asymptotic results for the two sequences of the associated…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree $\leq n$ in subintervals $\left [ a,b\right ] $ of the support of the underlying orthogonality measure…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

Every $n th$ order monic polynomial corresponds $n$-dimensional vector. If the given polynomial is stable that is all its roots lie in the open left half plane it is said to be Hurwitz polynomial and the corresponding vector is called…

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -\delta$ for some $0<\delta <1$. We show that for any $0 < \epsilon <1$, the value of $p(1)$ is determined within relative error $\epsilon$…

We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set of vectors of $\mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ of them include an orthogonal pair. We prove that for every prime…

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…

Define a subset of the complex plane to be a Rolle's domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolle's domain to be minimal if no proper subset is a Rolle's domain. In…

Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let q be a quadratic space over R on a free rank n R-module P such that the projective quadric q=0 is…

Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must…

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$ we ask the questions at which points its discriminant set can be considered as the graph of a function of all coefficients $a_j$ but one and how its subset of…