Related papers: Singular factors of rational plane curves
We prove the conjecture of Chen, Wang and Liu in [8] concerning how to calculate the parameter values corresponding to all the singu- larities, including the infinitely near singularities, of rational planar curves from the Smith normal…
Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from…
We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal…
It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve.…
In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…
Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal…
We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.
We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this…
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…
Given a curve $C$ over a number field $K$ equipped with the action of a finite group $G$ by $K$-automorphisms, one obtains a factorisation of $L(C,s)$ into a product of $L$-functions of `motivic pieces of curves' associated to irreducible…
We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve.…
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…
We study the matrix factorizations defined by the generic plane projections of a curve singularity of $\mathbb{C}^3$. On the other hand, given a plane curve singularity $Y\subset \mathbb{C}^2$ we study the family of matrix factorizations…
We use the method of the tensor product graph to construct rational (Yangian invariant) solutions of the Yang-Baxter equation in fundamental representations of $c_n$ and thence the full set of $c_n$-invariant factorized $S$-matrices. Brief…
We give an overview of the existing algorithms to compute nonunique factorization invariants in finitely generated monoids.
New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…
In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…
We enumerate factorisations of the complete graph into spanning regular graphs in several cases, including when the degrees of all the factors except for one or two are small. The resulting asymptotic behaviour is seen to generalise the…