3 research outputs found
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Shadows and intersections: stability and new proofs
We give a short new proof of a version of the Kruskal-Katona theorem due to
Lov\'asz. Our method can be extended to a stability result, describing the
approximate structure of configurations that are close to being extremal, which
answers a question of Mubayi. This in turn leads to another combinatorial proof
of a stability theorem for intersecting families, which was originally obtained
by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi
by means of a purely combinatorial result of Frankl. We also give an algebraic
perspective on these problems, giving yet another proof of intersection
stability that relies on expansion of a certain Cayley graph of the symmetric
group, and an algebraic generalisation of Lov\'asz's theorem that answers a
question of Frankl and Tokushige.Comment: 18 page