Related papers: Special points on intersections of hypersurfaces

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…

We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a…

In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is…

We consider resolutions of codimension-two enhanced singularities from $SO(12)$ to $E_7$ and from $E_7$ to $E_8$ in six-dimensional F-theory, where a half-hypermultiplet arises for generic complex structures achieving them. The exceptional…

We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the…

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…

We show that the term `superdifferential equation' has been employed in the literature to refer to different types of differential equations with even and odd variables. It is justified on physical and mathematical grounds that a subclass…

We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the…

We prove an effective bound for the degrees of generators of the algebraic de Rham cohomology of smooth affine hypersurfaces. In particular, we show that the de Rham cohomology H_dR^p(X) of a smooth hypersurface X of degree d in C^n can be…

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of…

We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper…

This paper develops the notion of superquadracity defined by L.Birbrair and M.Denkowski for subsets of R^n. In this regard, the main theorem of the paper establishes the relation between the superquadracity and non-empty intersection of the…

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…

We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…

Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the…

GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex…

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…

We present a novel feasibility criteria for the finite intersection of convex sets given by inequalities. This criteria allows us to easily assert the feasibility by analyzing the unconstrained minimum of a speci?fic convex function, that…

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…