811 research outputs found
On the number of ordinary conics
We prove a lower bound on the number of ordinary conics determined by a
finite point set in . An ordinary conic for a subset of
is a conic that is determined by five points of , and
contains no other points of . Wiseman and Wilson proved the
Sylvester-Gallai-type statement that if a finite point set is not contained in
a conic, then it determines at least one ordinary conic. We give a simpler
proof of their result and then combine it with a result of Green and Tao to
prove our main result: If is not contained in a conic and has at most
points on a line, then determines ordinary conics.
We also give a construction, based on the group structure of elliptic curves,
that shows that the exponent in our bound is best possible
On sets defining few ordinary lines
Let P be a set of n points in the plane, not all on a line. We show that if n
is large then there are at least n/2 ordinary lines, that is to say lines
passing through exactly two points of P. This confirms, for large n, a
conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for
this problem, as well as all sets having fewer than n - C ordinary lines for
some absolute constant C. We also solve, for large n, the "orchard-planting
problem", which asks for the maximum number of lines through exactly 3 points
of P. Underlying these results is a structure theorem which states that if P
has at most Kn ordinary lines then all but O(K) points of P lie on a cubic
curve, if n is sufficiently large depending on K.Comment: 72 pages, 16 figures. Third version prepared to take account of
suggestions made in a detailed referee repor
Finding an ordinary conic and an ordinary hyperplane
Given a finite set of non-collinear points in the plane, there exists a line
that passes through exactly two points. Such a line is called an ordinary line.
An efficient algorithm for computing such a line was proposed by Mukhopadhyay
et al. In this note we extend this result in two directions. We first show how
to use this algorithm to compute an ordinary conic, that is, a conic passing
through exactly five points, assuming that all the points do not lie on the
same conic. Both our proofs of existence and the consequent algorithms are
simpler than previous ones. We next show how to compute an ordinary hyperplane
in three and higher dimensions.Comment: 7 pages, 2 figure
Stratification of the fourth secant variety of Veronese variety via the symmetric rank
If is a projective non degenerate variety, the
-rank of a point is defined to be the minimum integer
such that belongs to the span of points of . We describe the
complete stratification of the fourth secant variety of any Veronese variety
via the -rank. This result has an equivalent translation in terms both
of symmetric tensors and homogeneous polynomials. It allows to classify all the
possible integers that can occur in the minimal decomposition of either a
symmetric tensor or a homogeneous polynomial of -border rank 4 (i.e.
contained in the fourth secant variety) as a linear combination of either
completely decomposable tensors or powers of linear forms respectively.Comment: In Press: Advances in Pure and Applied Mathematic
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