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Concept# Algebraic curve

Summary

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can tak

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Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer $n\ge 3$, we define $e(n)$ to be the minimum positive integer such that any set of $e(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $e(n) \le {2n-4 \choose n-2}+1$ and later in 1961, they obtained the lower bound $2^{n-2}+1 \le e(n)$, which they conjectured to be optimal. We prove that $e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2$. In a recent breakthrough, Suk proved that $e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}$. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let $c(n)$ (and $c'(n)$) denote the smallest positive integer $N$ such that any family of $N$ pairwise disjoint convex bodies in general position (resp., $N$ convex bodies in general position, any pair of which share at most two boundary points) has an $n$ members in convex position. We show that $c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}$. Given a point set $P$ in the plane, an ordinary circle for $P$ is defined as a circle containing exactly three points of $P$. We prove that any set of $n$ points in the plane, not all on a line or a circle, determines at least $\frac{1}{4}n^2-O(n)$ ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large $n$, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve $C\subset \mathbb{C}^2$ of degree $d$ and two finite sets $A,B\subset \mathbb{C}$, we have $|C\cap (A\times B)|=O_d(|A|+|B|)$. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection $|X\cap (P\times Q)|$ for a variety $X\subset \mathbb{C}^4$ and finite sets $P,Q\subset \mathbb{C}^2$. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that $\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}$ for any small set $A\subset \mathbb{F}_p$ and quadratic non-degenerate polynomial $f(x, y)\in \mathbb{F}_p[x, y]$. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term $\max \{ |A+A|, |A \cdot A |\}$. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on $\max\{ |A+B|, |f(B,C)|\}$, for a specific type of function $f$. Finally, we prove that the number of distinct cubic distances generated by any small set $A\times A\subset \mathbb{F}_p^2$ is $\Omega(|A|^{8/7})$, which improves a result of Yazici et al.

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.