Resultants and Singularities of Parametric Curves

We compute the $\delta$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic algebraic spatial curve projections). Our approach is based on some new results on zero loci of Schur polynomials, on transversality properties of maps defined by sparse polynomials, and on a new refinement of the notion of tropicalization ...

We correct a mistake in an earlier paper and give a complete classification of singular varieties having an extremal secant line.

When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interes...

We consider the interplay of point counts, singular cohomology, \'etale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for ze...

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 curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain 0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which encode all informat...

In this paper we present a constructive method to characterize ideals of the local ring $\mathscr{O}_{\mathbb{C}^n,0}$ of germs of holomorphic functions at $0\in\mathbb{C}^n$ which arise as the moduli ideal $\langle f,\mathfrak{m}\, j(f)\rangle$, for some $f\in\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^n,0}$. A consequence of our characterization is an effective solution to a problem dating back to the 1980's, called the Reconstruction Problem of the hypersurface singularity ...

In the present paper, we prove the existence of universal polynomials which express multi-singularity loci classes of prescribed types for proper morphisms between smooth schemes over an algebraically closed field of characteristic zero -- we call them Thom polynomials for multi-singularity types of maps. It has been referred to as the Thom-Kazarian principle and unsolved for a long time. This result solidifies the foundation for a general enumerative theory of singularities ...

In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined over a large class of coefficient rings by means of the resultant. Many formal properties and computational rules are provided and the geometric interpretation of the discriminant is investigated over a general coefficient ring, typically a ...

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map f...