Related papers: Algebraic Multiplicity and the Poincar\'{e} Proble…
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…
The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is…
This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincar\'{e} series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincar\'e series was…
This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate…
We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the…
We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes…
Let $C$ be a generic complex plane plane curve with a given Newton polygon $P$. We compute the number of its inflection points and bitangents (equivalently, the number of singularities of the projectively dual curve $C^\vee$). We also prove…
We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit…
We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds…
In this note we revisit the problem of determining combinatorially the multiplicity at the origin of a toric curve. In addition, we give the exact value of the regularity index of that point for plane toric curves and effective bounds for…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
We compute the A-infinity structure on the self-Ext algebra of the vector bundle $G$ over an elliptic curve of the form $G=\bigoplus_{i=1}^r P_i\oplus \bigoplus_{j=1}^s L_j$, where $(P_i)$ and $(L_j)$ are line bundles of degrees 0 and 1,…
Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…
An explicit upper bound is derived for the modulus of divided difference for a smooth(not necessarily analytic) function defined on a smooth Jordan arc (or a smooth Jordan curve) in the complex plane. As an immediate application, an error…
As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes…
For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.
Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…
For an arbitrary representation $\rho$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $\rho$. Among other interesting properties, these numbers provide lower…