Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular ho-motopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained… (More)
— In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input–multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove… (More)
Smale's 17th Problem asks " Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm? " We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a… (More)
—In this paper, we propose a test for checking the feasibility of linear interference alignment (IA) for multiple-input multiple-output (MIMO) channels with constant coefficients for any number of users, antennas and streams per user. We consider the compact complex manifold formed by those channels, pre-coders and decoders that satisfy the polynomial IA… (More)
We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input. We show that the method can also be used to approximate several or all the solutions of non–degenerate systems,… (More)
We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given two pairs (f i , ζ i), i = 1, 2, there exists a short path joining them such that the complexity of following the path is bounded by the logarithm of the condition number of the problems.
We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional varieties. We also state an upper bound for the volume of… (More)
In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input multiple-output (MIMO) channels with constant coefficients for any number of users, antennas and streams per user. We combine algebraic geometry techniques with differential topology ones and prove much stronger results than those previously published on this… (More)