6 research outputs found
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in R2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is (Formula presented.). We establish the slightly weaker bound (Formula presented.) on the number of incidences between m points and n (complex) algebraic curves in C2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over C. © 2017 Jånos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelber
Schwartz-Zippel bounds for two-dimensional products
Schwartz-Zippel bounds for two-dimensional products, Discrete Analysis 2017:20,
A famous open problem in combinatorial geometry is ErdĆs's unit-distances problem, which asks the following: given a subset of size , how many pairs can there be with ? (Here is the usual Euclidean distance.) The best-known upper bound, due to Spencer, SzemerĂ©di and Trotter, is , but it is conjectured that the true bound is for every . (By putting all the points along a line, one can obtain a lower bound of with ease.) This problem is closely related to ErdĆs's distinct-distances problem, which was spectacularly solved by Guth and Katz, who showed that every set of size in must give rise to at least distinct distances. This would be a consequence of a positive answer to the unit-distances problem, since there are pairs of points and each distance would occur at most times.
An equally famous theorem in combinatorial geometry is the Szemerédi-Trotter theorem, which asserts that amongst any points and lines in , the number of incidences (that is, pairs where is one of the points, is one of the lines, and is on ) is at most .
This paper concerns a simultaneous generalization of the Szemerédi-Trotter theorem and the upper bound above for the unit-distances problem. The connection is that both problems can be viewed as giving upper bounds for the size of the intersection of a variety with a Cartesian product of two finite subsets of the plane. In the case of the unit-distances problem, the two subsets are equal and the variety is the zero set of the function . The intersection of this zero set with the Cartesian product is the set of all pairs such that and , so the size of the intersection is the number of unit distances in . As for the Szemerédi-Trotter theorem, if we let be the set of points, given by their Cartesian coordinates, and be the set of lines, associating the line with the pair , then the point belongs to the line if and only if , so the number of incidences is the size of the intersection of with the zero set of the polynomial .
This observation makes it tempting to conjecture that for every non-zero 4-variable polynomial and every pair of subsets of of size there is an upper bound of on the intersection of with the zero set of . However, this is easily seen to be false. If has a formula of the form , then is contained in the zero set of if vanishes on and vanishes on . So some condition is needed on the polynomial for the conjecture to have a chance of being correct. The main result of this paper is essentially that the above source of examples is the only one: if a polynomial in four complex variables cannot be written in this way, then there is an upper bound of for any for the intersection of its zero set with a Cartesian product of two subsets of of size . It is not clear whether this can be improved to an upper bound of -- hence the word "essentially" above.
The title of the paper comes from the fact that this result can be viewed as a generalization of a special case of the Schwartz-Zippel lemma, which concerns products of subsets of rather than subsets of . The proof depends on a two-dimensional generalization of a special case of yet another central result in combinatorial geometry, Alon's combinatorial Nullstellensatz. The paper also contains results about varieties of dimensions 1 and 2: for these the authors obtain a linear upper bound for the size of the intersection.
The Szemerédi-Trotter theorem is known to give a tight bound, so in general the main result of the paper cannot be improved. However, this does not rule out improvements for specific polynomials. In the light of the unit-distances problem, it would naturally be of great interest to understand which polynomials might give rise to upper bounds that are better than . One of the merits of this paper is that it focuses our attention on this question