Related papers: S-curves in Polynomial External Fields

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…

Let $f:\mathbb{K}^n\rightarrow\mathbb{K}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In arXiv:1411.5011 we proved that if $\mathbb{K}$ is the field of complex or real numbers, then the set $S_f$ of points…

The van der Geer-van der Vlugt curves are Artin-Schreier coverings of the affine line defined by linearized polynomials over finite fields. We give several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound…

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots)…

We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this…

We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is…

Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of V consisting of the polynomials that are bounded on S. We introduce the notion of…

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More…

Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some…

We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with…

We give a combinatorial proof of an unpublished result of E. Klarreich: The Gromov boundary of the complex of curves of a non-exceptional oriented surface S of finite type can naturally be identified with the space of minimal geodesic…

We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from…

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all…

In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we…

In this note we study a minimization problem for a vector of measures subject to a prescribed interaction matrix in the presence of external potentials. The conductors are allowed to have zero distance from each other but the external…

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any…

Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces S with singular points of prescribed topological types S_1,...,S_r. There are necessary conditions for the existence of the type \sum_{i=1}^r…

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results,…

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrized by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at…

Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, G. Minervini proved in his PhD thesis [17], among other things, the Harvey-Lawson Diagonal Theorem but without the restrictive tameness condition for Morse…