Related papers: On a stratification defined by real roots of polyn…

We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper.…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has…

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…

A classification and a detailed geometric description are given for smooth $n$-dimensional subvarieties $X\subset{\mathbb P}^{2n-1}$ containing a family of effective divisors each of them spanning a linear ${\mathbb P}^n$ of ${\mathbb…

We specify the canonical stratifications satisfying respectively Whitney (a)-regularity, Whitney (b)-regularity, Kuo-Verdier (w)-regularity, and Mostowski (L)-regularity for the family of surfaces y^a = t^b x^c + x^d, where a, b, c, d are…

We study several properties of the symmetric power $S^mX$ of a smooth variety $X$. We describe the Picard and divisor class groups of $S^mX$ when $X$ is projective. We give a complete description of the stratification of $S^mX$ by iterated…

In this article, we establish necessary and sufficient conditions for a polynomial of degree $n$ to have exactly $n$ real roots. A complete study of polynomials of degree five is carried out. The results are compared with those obtained…

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for…

The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…

We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in…

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

We give the stratification by the symmetric tensor rank of all degree $d \ge 9$ homogeneous polynomials with border rank $5$ and which depend essentially on at least 5 variables, extending previous works (A. Bernardi, A. Gimigliano, M.…

The degree polynomial of a multigraph $G$ is given by $\sum _{v \in V(G)} x^{\mbox{deg}(v)}$. We investigate here properties of the roots of such polynomials. In addition to examining the roots for some families of graphs with few and many…

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex…

Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the…

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n\xi_kz^k$ with i.i.d. coefficients $\xi_0,\ldots,\xi_n$…

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices. Specifically, we prove that if $X_1,\dots ,X_n$ are…