Related papers: Adjunction inequality for real algebraic curves

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two…

For any natural $d \ge k \ge 2$ we calculate the cohomology groups of the space of homogeneous polynomials $R^2 \to R$ of degree $d$, which do not vanish with multiplicity $\ge k$ on real lines. For $k=2$ this problem provides the simplest…

This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many…

In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[…

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…

We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The…

Let T be a topology on the finite set Xn. We consider the open polynomial associated with the topology T. Its coefficients are the cardinality of open sets of size j=0,...,n. J. Brown [4] asked when this polynomial has only real zeros. We…

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to…

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…

We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…

We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…

The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $\Sigma$. The fundamental second kind differential is a symmetric $1\otimes 1$ form on $\Sigma\times \Sigma$ that has a double pole at coinciding points…

We study the number of real zeros of trigonometric polynomials in a period and the number of zeros of self-reciprocal algebraic polynomials on the unit circle under the assumption that their coefficients are in a fixed finite set of real…

For a planar graph with a given f-vector $(f_{0}, f_{1}, f_{2}),$ we introduce a cubic polynomial whose coefficients depend on the f-vector. The planar graph is said to be real if all the roots of the corresponding polynomial are real. Thus…

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the…