Related papers: On the intersection problem for linear sets in the…
If new particles are discovered, it will be important to determine if they are the supersymmetric partners of standard model bosons and fermions. Supersymmetry predicts relations among the couplings and masses of these particles. We discuss…
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has…
An $n$-correct set $\mathcal{X}$ in the plane is a set of nodes admitting unique interpolation with bivariate polynomials of total degree at most $n$. A $k$-node line is a line passing through exactly $k$ nodes of $\mathcal{X}.$ A line can…
Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have…
In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.
A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two…
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops…
We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in…
The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods…
We propose algorithms and software for computing projections onto the intersection of multiple convex and non-convex constraint sets. The software package, called SetIntersectionProjection, is intended for the regularization of inverse…
We formulate problems of tight closure theory in terms of projective bundles and subbundles. This provides a geometric interpretation of such problems and allows us to apply intersection theory to them. This yields new results concerning…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…
It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear…
In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon…
We study a general method of the field intersection problem of generic polynomials over an arbitrary field $k$ via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using…
We study the problem of learning permutation invariant representations that can capture "flexible" notions of containment. We formalize this problem via a measure theoretic definition of multisets, and obtain a theoretically-motivated…
Let $U$ be a set of polynomials of degree at most $k$ over $\mathbb{F}_q$, the finite field of $q$ elements. Assume that $U$ is an intersecting family, that is, the graphs of any two of the polynomials in $U$ share a common point.…
We investigate the maximum number of intersections between two polygons with p and q vertices, respectively, in the plane. The cases where p or q is even or the polygons do not have to be simple are quite easy and already known, but when p…
Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In contrast, there are few results on analogous problems…
We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has…