An upper bound for the minimum diameter of integral point sets

We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of simultaneous flips that are needed to transform a triangulation into another. Our results hold for triangulations of point sets and for polygons.

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Counting Triangulations of Planar Point Sets

The Electronic Journal of Combinatorics ◽

10.37236/557 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 25

Author(s):

Micha Sharir ◽

Adam Sheffer

Keyword(s):

Upper Bound ◽

Planar Graphs ◽

Spanning Trees ◽

Upper Bounds ◽

Line Graphs ◽

Point Sets ◽

Planar Point ◽

Straight Line ◽

Point Set ◽

Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

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A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

The Electronic Journal of Combinatorics ◽

10.37236/484 ◽

2010 ◽

Vol 17 (1) ◽

Author(s):

Ondřej Bílka ◽

Kevin Buchin ◽

Radoslav Fulek ◽

Masashi Kiyomi ◽

Yoshio Okamoto ◽

...

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Minkowski Sum ◽

Independent Subset ◽

Point Sets ◽

Planar Point ◽

Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

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ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000719 ◽

2009 ◽

Vol 81 (1) ◽

pp. 114-120 ◽

Cited By ~ 3

Author(s):

IGOR E. SHPARLINSKI

Keyword(s):

Finite Field ◽

Finite Fields ◽

Upper Bound ◽

Dimensional Space ◽

Real Space ◽

Acute Angle ◽

Vector Spaces ◽

Point Sets ◽

Computation of Mordell–Weil bases for ordinary elliptic curves in characteristic two

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000175 ◽

2014 ◽

Vol 17 (A) ◽

pp. 1-13

Author(s):

G. Moehlmann

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Upper Bound ◽

Duality Theorem ◽

Homogeneous Spaces ◽

Integral Point ◽

Selmer Groups ◽

Global Function ◽

Global Function Fields ◽

Characteristic Two

AbstractIn this paper we consider ordinary elliptic curves over global function fields of characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an $S$-integral point on $A$. To illustrate our methods, a detailed example is presented.

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