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- Carlos Beltrán, Luis M. Pardo
- Foundations of Computational Mathematics
- 2011

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)

- Oscar Gonzalez, Carlos Beltrán, Ignacio Santamaría
- IEEE Transactions on Information Theory
- 2014

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… (More)

In the series of papers [SS93a, SS93b, SS93c, SS94, SS96], Shub and Smale defined and studied in depth a homotopy method for solving systems of polynomial equations. Some articles preceding this new treatment were [Kan49, Sma86, Ren87, Kim89, Shu93]. Other authors have also treated this approach in [Mal94, Yak95, Ded97, BCSS98, Ded01, Ded06, MR02] and more… (More)

- Carlos Beltrán, Anton Leykin
- Experimental Mathematics
- 2012

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… (More)

- Carlos Beltrán, Anton Leykin
- Foundations of Computational Mathematics
- 2013

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. This paper deals with the problem of distributing points… (More)

- Carlos Beltrán, Luis M. Pardo
- Foundations of Computational Mathematics
- 2007

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)

- Carlos Beltrán
- Numerische Mathematik
- 2011

- Oscar Gonzalez, Ignacio Santamaría, Carlos Beltrán
- 2012 IEEE International Symposium on Information…
- 2012

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)

- Carlos Beltrán, Luis M. Pardo
- Foundations of Computational Mathematics
- 2008

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)