相关论文: On modular inequalities for plane projective curve…
In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor-L\^e Fibration and the equivalence between them.…
We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve $C$ with the Mordell--Weil rank of certain isotrivial families of jacobians over $\mathbf{P}^2$ of discriminant $C$. In the second part…
The twisted Alexander polynomials of a space, associated to a linear representation $\sigma$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they…
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…
Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor…
First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function…
We have generalized the Edwards' method of collective description of dense polymer systems in terms of effective potentials to polymer blends in the presence of a surface. With this method we have studied conformationally asymmetric…
We study tori attached to the fundamental groups of plane curves with arbitrary singularities. These tori provide complete information about homology of finite abelian covers of the plane branched along the curve. We calculate these tori in…
We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called…
The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite…
We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links,…
Singularities of plane into plane mappings described by parabolic two-component systems of quasi-liner partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach…
The twisted Alexander polynomial of a knot is defined associated to a linear representation of the knot group. If there exists a surjective homomorphism of a knot group onto a finite group, then we obtain a representation of the knot group…
We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $\theta$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show…
Planar curves with periodically varying curvature arise in the natural sciences as the result of a wide variety of periodic processes. The total curvature of a periodic arc in such curves constrains their symmetry. It is shown how the total…
A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and…
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using…
A famous theorem of Weyl states that if $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain…