8 research outputs found
Average volume, curvatures, and Euler characteristic of random real algebraic varieties
We determine the expected curvature polynomial of random real projective
varieties given as the zero set of independent random polynomials with Gaussian
distribution, whose distribution is invariant under the action of the
orthogonal group. In particular, the expected Euler characteristic of such
random real projective varieties is found. This considerably extends previously
known results on the number of roots, the volume, and the Euler characteristic
of the solution set of random polynomial equationsComment: 38 pages. Version 2: corrected typos, changed some notation, rewrote
proof of Theorem 5.
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