Related papers: Quadratic vector fields with invariant algebraic c…

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in…

In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.

In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.

We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.

We study the plane automorphisms given by polynomials with certain degree decompositions.

The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ as an invariant set. We first characterize…

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic…

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.

For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of…

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and non degenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central…

A classification of the global structure of monic and centered one-variable complex polynomial vector fields is presented.

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

We study the sheaves of logarithmic vector fields along smooth cubic curves in the projective plane, and prove a Torelli-type theorem in the sense of Dolgachev-Kapranov for those with non-vanishing j-invariants.

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

For all positive integers $k$ and $N$ we prove that there are infinitely many totally real multiquadratic fields $K$ of degree $2^k$ over $\mathbb Q$ such that each universal quadratic form over $K$ has at least $N$ variables.

We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.

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…