Counting joints with multiplicities

Iliopoulou, Marina

doi:10.1112/plms/pds052

Cited by 9 publications

(14 citation statements)

References 19 publications

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“…Let us mention here that (7) has been recently proved for generic joints in any field setting by Hablicsek (in [17]). Moreover, we have shown (7) in R 3 in [20], without the genericity hypothesis, but with an extra logarithmic factor of L in its right-hand side, which we believe is not necessary (the logarithmic factor of 1/δ on the right-hand side of (4) reflects the fact that two tubes may intersect a lot; two lines, on the other hand, intersect at at most one point).…”

Section: Lnmentioning

confidence: 78%

Incidence Bounds on Multijoints and Generic Joints

Iliopoulou

2015

Discrete Comput Geom

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Abstract.A point x ∈ F n is a joint formed by a finite collection L of lines in F n if there exist at least n lines in L through x that span F n . It is known that there are n |L| n n−1 joints formed by L. We say that a point x ∈ F n is a multijoint formed by the finite collections L1, . . . , Ln of lines in F n if there exist at least n lines through x, one from each collection, spanning F n . We show that there are n (|L1| · · · |Ln|) 1 n−1 such points for any field F and n = 3, as well as for F = R and any n ≥ 3.Moreover, we say that a point x ∈ F n is a generic joint formed by a finite collection L of lines in F n if each n lines of L through x form a joint there. We show that, for F = R and any n ≥ 3, there areThis result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in R 3 in [16] by Guth and Katz.Finally, we generalise our results in R n to the case of multijoints and generic joints formed by real algebraic curves.

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Section: Lnmentioning

confidence: 78%

Incidence Bounds on Multijoints and Generic Joints

Iliopoulou

2015

Discrete Comput Geom

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“…The following holds (see [14] for a detailed proof). On a different subject, it is known (see [3] or [1,Chapter 5]) that each real semialgebraic set (i.e., any set of the form {x ∈ R n : P (x) = 0 and Q(x) > 0, ∀ Q ∈ Q}, where P ∈ R[x 1 , .…”

Section: From Lines To Curvesmentioning

confidence: 79%

“…Using Corollary 4.2 and the above-mentioned characterisation of the dimension of a variety via term orders on sets of monomials, we have proved the following in [14]. Note that, by the above, the smallest complex algebraic curve containing a real algebraic curve is the union of the irreducible complex algebraic curves, each of which contains an irreducible component of the real algebraic curve.…”

Section: From Lines To Curvesmentioning

confidence: 81%

“…Now, in analogy to the notion of critical lines, we say that a curve contained in the zero set Z of a polynomial in R[x, y, z] is a critical curve if all the points of the curve are critical points of Z. We have shown in[14] that the following holds.…”

mentioning

confidence: 82%

“…In the light of this, Carbery has conjectured that, if, for each joint x ∈ J, the multiplicity M (x) of x is the number of n-tuples of linearly independent lines of L passing through x, then x∈J M (x) 1/(n−1) L n/(n−1) ; ( 2 ) in fact, we have shown (2) for n = 3 in [14]. Now, if L 1 , .…”

Section: Introductionmentioning

confidence: 90%

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Counting multijoints

Iliopoulou

2015

Journal of Combinatorial Theory, Series A

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Let L 1 , L 2 , L 3 be finite collections of L 1 , L 2 , L 3 , respectively, lines in R 3 , and J(L 1 , L 2 , L 3 ) the set of multijoints formed by them, i.e. the set of points x ∈ R 3 , each of which lies in at least one line l i ∈ L i , for all i = 1, 2, 3, such that the directions of l 1 , l 2 and l 3 span R 3 . We prove here that |J(L 1 , L 2 , L 3 )| (L 1 L 2 L 3 ) 1/2 (which was previously known when L 1 , L 2 and L 3 are roughly the same), 1 and we extend our results to multijoints formed by real algebraic curves in R 3 of uniformly bounded degree, as well as by curves in R 3 parametrised by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, as well as a discrete analogue of the endpoint multilinear Kakeya problem. address: M.Iliopoulou@bham.ac.uk. 1 Any expression of the form A B or A = O(B) means that there exists a non-negative constant M , depending only on the dimension, such that A ≤ M · B. Any expression of the form A b 1 ,...,b m B or A = O b 1 ,...,b m (B) means that there exists a constant M b 1 ,...,b m , depending only on the dimension and b 1 , . . . , b m , such that A M b 1 ,...,b m · B. Moreover, we write A B or A = Ω(B) if B A, and we write A b 1 ,...,b m B or A = Ω b 1 ,...,b m (B) if B b 1 ,...,b m A. Finally, A ∼ B means that A B and A B, while A ∼ b 1 ,...,b m B means that A b 1 ,...,b m B and A b 1 ,...,b m B. http://dx.

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Joints of Varieties

Tidor

Zhao

2022

Geom. Funct. Anal.

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We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that N planes (2-flats) in 6 dimensions (over any field) have $$O(N^{3/2})$$ O ( N 3 / 2 ) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of N planes is replaced by a set of 2-dimensional algebraic varieties of total degree N, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery’s conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.

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Counting joints with multiplicities

Cited by 9 publications

References 19 publications

Incidence Bounds on Multijoints and Generic Joints

Incidence Bounds on Multijoints and Generic Joints

Counting multijoints

Joints of Varieties

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