Fetching the paper…

Reading the bibliography…

2014

Sylvester-Gallai for Arrangements of Subspaces

Dvir, Zeev, Hu, Guangda

Original
PDF
Post on X

Understand

In this work we study arrangements of $k$-dimensional subspaces $V_1,\ldots,V_n \subset \mathbb{C}^\ell$.

Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$ contained in a $2k$-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on $k$ (and not on $n$).
The theorem holds under the assumption that $V_a \cap V_b = \{0\}$ for every pair (otherwise it is false).
This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the $k=1$ case.

Sylvester-Gallai for Arrangements of Subspaces · Around

Built on

Uber vielseite der projektive ebene
E. Melchior · 1940
Earlier work this paper cites.
Problems for solution: 4065-4069
P. Erdös, Richard Bellman, H. S. Wall, James Singer, and V. Thébault · 1943
Earlier work this paper cites.
A generalization of a theorem of Sylvester on the lines determined by a finite point set
Sten Hansen · 1965
Earlier work this paper cites.
A resolution of the Sylvester-Gallai problem of J.-P. Serre
L. M. Kelly · 1986
Earlier work this paper cites.
A survey of Sylvester’s problem and its generalizations
P. Borwein and W. O. J. Moser · 1990
Earlier work this paper cites.

Similar

On a reverse form of the Brascamp-Lieb inequality
Franck Barthe · 1998
Cited alongside, same era.
Sylvester-Gallai theorems for complex numbers and quaternions
Noam Elkies, Lou M Pretorius, and Konrad J Swanepoel · 2006
Cited alongside, same era.
Linear algebra and its applications
Peter D. Lax · 2007
Cited alongside, same era.
Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes
Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson · 2011
Cited alongside, same era.
Mathematical question 11851
J. J. Sylvester
Cited in the paper.

Then

On matrix rigidity and locally self-correctable codes
Zeev Dvir · 2011
Later among the works it cites.
Fractional Sylvester-Gallai theorems
Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff · 2013
Later among the works it cites.
Breaking the quadratic barrier for 3-LCC’s over the reals
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson · 2014
Closest in time.
Improved rank bounds for design matrices and a new proof of Kelly’s theorem
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson · 2014
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…