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A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$…

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…

For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of…

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of…

We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real…

Let $X\subset\Bbb C^n$ be an affine variety and $f:X\to\Bbb C^m$ be the restriction to $X$ of a polynomial map $\Bbb C^n\to\Bbb C^m$. In this paper, we construct an affine Whitney stratification of $X$. The set $K(f)$ of stratified…

Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given.…

We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the…

We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is…

For maps of one complex variable, $f$, given as the sum of a degree $n$ power map and a degree $d$ polynomial, we provide necessary and sufficient conditions that the geometric limit as $n$ approaches infinity of the set of points that…

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construc-tion uses a single linear differential form defined from the…

We start from a parametrized system of $d$ generalized polynomial equations (with real exponents) for $d$ positive variables, involving $n$ generalized monomials with $n$ positive parameters. Existence and uniqueness of a solution for all…

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials…

Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajb\'ok, Marino,…

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse…

Let X \subset R be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e.the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where…

We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is isolated if it is not in a family of effective degree $d$ divisors parametrized by…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…