Related papers: Solvable points on intersections of quadrics, cubi…

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which…

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano…

Let $X^o=\mathbb P^3\setminus D$ where $D$ is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface $V$ and a plane…

We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of…

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

In this article we show that non-singular quadrics and non-singular Hermitian varieties are completely characterized by their intersection numbers with respect to hyperplanes and spaces of codimension 2. This strongly generalizes a result…

This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…

Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of…

This paper was written in 1982. Ideas and methods of "Clemens C.H., Griffiths Ph. The intermediate Jacobian of a cubic threefold" are applied to a Fano threefold X of genus 6 -- intersection of Grassmann sixfold with two hyperplanes and a…

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…

In the calculation of thermodynamic properties and three dimensional structures of macromolecules, such as proteins, it is important to have a good algorithm for computing solvent accessible surface area of macromolecules. Here we propose a…

We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…

For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.

We survey a number of results on the counting of points on hypersurfaces defined over finite fields. We also investigate when one can be guaranteed a non-singular point on a projective hypersurface and give a condition on the cardinality of…

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by…

Hrushovski's generalization and application of [Jouanolou, "Hypersurfaces solutions d'une \'equation de Pfaff analytique", Mathematische Annalen, 232 (3):239--245, 1978] is here refined and extended to the partial differential setting with…

We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof…