2,298 research outputs found
Fixing numbers for matroids
Motivated by work in graph theory, we define the fixing number for a matroid.
We give upper and lower bounds for fixing numbers for a general matroid in
terms of the size and maximum orbit size (under the action of the matroid
automorphism group). We prove the fixing numbers for the cycle matroid and
bicircular matroid associated with 3-connected graphs are identical. Many of
these results have interpretations through permutation groups, and we make this
connection explicit.Comment: This is a major revision of a previous versio
Tropical Quadrics Through Three Points
We tropicalize the rational map that takes triples of points in the
projective plane to the plane of quadrics passing through these points. The
image of its tropicalization is contained in the tropicalization of its image.
We identify these objects inside the tropical Grassmannian of planes in
projective 5-space, and we explore a small tropical Hilbert scheme.Comment: 9 pages, 3 figure
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
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