Carlos Beltrán

A Feasibility Test for Linear Interference Alignment in MIMO Channels With Constant Coefficients

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 a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class.The work of O. Gonzalez and I. Santamaria was supported by MICINN (Spanish Ministry for Science and Innovation) under grants TEC2010-19545-C04-03 (COSIMA), CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS) and FPU grant AP2009-1105. Carlos Beltran was partially supported by the MICINN grant MTM2010-16051.

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

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 (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.

Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

Smale’s 17th Problem asks: “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?”. We give a positive answer to this question. Namely, we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations f : Cn −→ Cn, performing a number of arithmetic operations which is polynomial in the size of the input, on the average.

Complexity of Bezout’s Theorem VII: Distance Estimates in the Condition Metric

Abstract We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given the two pairs (fi,ζi), i=1,2, there exist 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.

CERTIFIED NUMERICAL HOMOTOPY TRACKING

Given a homotopy connecting two polynomial systems, we provide a rigorous algorithm for tracking a regular homotopy 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 with an implementation in the numerical algebraic geometry package Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.

On Smale's 17th Problem: A Probabilistic Positive Solution

Smales 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 partial positive solution to Smales 17th Problem.

Robust Certified Numerical Homotopy Tracking

We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.

Protein arginine methyl transferases-3 and -5 increase cell surface expression of cardiac sodium channel

The α‐subunit of the cardiac voltage‐gated sodium channel (NaV1.5) plays a central role in cardiomyocyte excitability. We have recently reported that NaV1.5 is post‐translationally modified by arginine methylation. Here, we aimed to identify the enzymes that methylate NaV1.5, and to describe the role of arginine methylation on NaV1.5 function. Our results show that protein arginine methyl transferase (PRMT)‐3 and ‐5 methylate NaV1.5 in vitro, interact with NaV1.5 in human embryonic kidney (HEK) cells, and increase NaV1.5 current density by enhancing NaV1.5 cell surface expression. Our observations are the first evidence of regulation of a voltage‐gated ion channel, including calcium, potassium, sodium and TRP channels, by arginine methylation.

A general test to check the feasibility of linear interference alignment

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 equations (the so-called solution variety), and study its projection onto the input space formed by the interference channels. When the derivative of this projection is surjective, namely when the tangent space of the solution variety is projected into the whole tangent space of the inputs space, the linear alignment problem is feasible; otherwise is infeasible. Building on these results, a general feasibility test, which amounts to check whether a given matrix is full-rank or not, is proposed.

Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials

We prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere. That is, roots of random polynomials provide a fairly good approximation to elliptic Fekete points.