Related papers: Bounds of incidences between points and algebraic …

We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level $\ell$" points, in the sense introduced by Arnold-Shimura, given as a polynomial in the numbers of hypersurfaces…

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.

We show that the number of incidences between $m$ distinct points and $n$ distinct lines in ${\mathbb R}^4$ is $O\left(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n\right)$, for a suitable…

It is shown that under certain boundary conditions the virial theorem has to be modified. We analyze the origin of the extra term and compute it in particular examples. The Coulomb and harmonic oscillator with point interaction have been…

We establish a majorization-based theory for bounding observables of waves with varied coherence. For any measurement, exact bounds are attained by the maximal and minimal elements in the set of input coherence spectra. The set's supremum…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable…

We study the minimal degrees and gonalities of curves on complete intersections. We prove a classical conjecture which asserts that the degree of any curve on a general complete intersection $X \subseteq \mathbb{P}^N$ cut out by polynomials…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1…

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given…

We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different…

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square…

This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…

There are two purposes in this article. One is to present a criterion for the existence of a birational embedding into a projective plane with inner and outer Galois points for algebraic curves. Another is to classify plane curves of degree…

In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three…

We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a)…