1,984 research outputs found
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
Bounded negativity, Harbourne constants and transversal arrangements of curves
The Bounded Negativity Conjecture predicts that for every complex projective
surface there exists a number such that holds for
all reduced curves . For birational surfaces there have
been introduced certain invariants (Harbourne constants) relating to the effect
the numbers , and the complexity of the map . These invariants
have been studied previously when is the blowup of all singular points of
an arrangement of lines in , of conics and of cubics. In the
present note we extend these considerations to blowups of at
singular points of arrangements of curves of arbitrary degree . We also
considerably generalize and modify the approach witnessed so far and study
transversal arrangements of sufficiently positive curves on arbitrary surfaces
with the non-negative Kodaira dimension.Comment: This is the final version, incorporating the suggestions of the
referee, to appear in Annales de l'Institut Fourier Grenobl
Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance
Given two polygonal curves in the plane, there are many ways to define a
notion of similarity between them. One popular measure is the Fr\'echet
distance. Since it was proposed by Alt and Godau in 1992, many variants and
extensions have been studied. Nonetheless, even more than 20 years later, the
original algorithm by Alt and Godau for computing the Fr\'echet
distance remains the state of the art (here, denotes the number of edges on
each curve). This has led Helmut Alt to conjecture that the associated decision
problem is 3SUM-hard.
In recent work, Agarwal et al. show how to break the quadratic barrier for
the discrete version of the Fr\'echet distance, where one considers sequences
of points instead of polygonal curves. Building on their work, we give a
randomized algorithm to compute the Fr\'echet distance between two polygonal
curves in time on a pointer machine
and in time on a word RAM. Furthermore, we show that
there exists an algebraic decision tree for the decision problem of depth
, for some . We believe that this
reveals an intriguing new aspect of this well-studied problem. Finally, we show
how to obtain the first subquadratic algorithm for computing the weak Fr\'echet
distance on a word RAM.Comment: 34 pages, 15 figures. A preliminary version appeared in SODA 201
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