811 research outputs found

    On the number of ordinary conics

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    We prove a lower bound on the number of ordinary conics determined by a finite point set in R2\mathbb{R}^2. An ordinary conic for a subset SS of R2\mathbb{R}^2 is a conic that is determined by five points of SS, and contains no other points of SS. 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 SS is not contained in a conic and has at most c∣S∣c|S| points on a line, then SS determines Ωc(∣S∣4)\Omega_c(|S|^4) 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

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    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

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    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

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    If X⊂PnX\subset \mathbb{P}^n is a projective non degenerate variety, the XX-rank of a point P∈PnP\in \mathbb{P}^n is defined to be the minimum integer rr such that PP belongs to the span of rr points of XX. We describe the complete stratification of the fourth secant variety of any Veronese variety XX via the XX-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers rr that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of XX-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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