Improved method for finding optimal formulae for bilinear maps in a finite field

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae, in particular for the short product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we are able to prove that there is essentially only one optimal decomposition of the product of 3 x 2 by 2 x 3 matrices up to the action of some group of automorphisms

The Bounded L2 Curvature Conjecture

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear $so(3,1)$-valued Yang-Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of \textit{null structure} compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including $L^2$ error bounds which is carried out in [Sz1]-[Sz4], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in \cite{Sz5}.Comment: updated version taking into account the remarks of the refere

Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message-passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the asymptotic support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

International audienceWe indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in O(n(2q)^log * q (n)) where n corresponds to the extension degree, and log * q (n) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field F 3^57. Index Terms Multiplication algorithm, bilinear complexity, elliptic function field, interpolation on algebraic curve, finite field