A primal-dual formulation for certifiable computations in Schubert calculus

Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's \alpha-theory.Comment: 21 page

Interference alignment for the MIMO interference channel

We study vector space interference alignment for the MIMO interference channel with no time or frequency diversity, and no symbol extensions. We prove both necessary and sufficient conditions for alignment. In particular, we characterize the feasibility of alignment for the symmetric three-user channel where all users transmit along d dimensions, all transmitters have M antennas and all receivers have N antennas, as well as feasibility of alignment for the fully symmetric (M=N) channel with an arbitrary number of users. An implication of our results is that the total degrees of freedom available in a K-user interference channel, using only spatial diversity from the multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees of freedom shown to be possible by Cadambe and Jafar with arbitrarily large time or frequency diversity. Moving beyond the question of feasibility, we additionally discuss computation of the number of solutions using Schubert calculus in cases where there are a finite number of solutions.Comment: 16 pages, 7 figures, final submitted versio

Reality and Computation in Schubert Calculus

The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. When conjectured, it sparked interest in real osculating Schubert calculus, and computations played a large role in developing the surrounding theory. Our purpose is to uncover generalizations of the Mukhin-Tarasov-Varchenko Theorem, proving them when possible. We also improve the state of the art of computationally solving Schubert problems, allowing us to more effectively study ill-understood phenomena in Schubert calculus. We use supercomputers to methodically solve real osculating instances of Schubert problems. By studying over 300 million instances of over 700 Schubert problems, we amass data significant enough to reveal generalizations of the Mukhin-Tarasov- Varchenko Theorem and compelling enough to support our conjectures. Combining algebraic geometry and combinatorics, we prove some of these conjectures. To improve the efficiency of solving Schubert problems, we reformulate an instance of a Schubert problem as the solution set to a square system of equations in a higher- dimensional space. During our investigation, we found the number of real solutions to an instance of a symmetrically defined Schubert problem is congruent modulo four to the number of complex solutions. We proved this congruence, giving a generalization of the Mukhin-Tarasov-Varchenko Theorem and a new invariant in enumerative real algebraic geometry. We also discovered a family of Schubert problems whose number of real solutions to a real osculating instance has a lower bound depending only on the number of defining flags with real osculation points. We conclude that our method of computational investigation is effective for uncovering phenomena in enumerative real algebraic geometry. Furthermore, we point out that our square formulation for instances of Schubert problems may facilitate future experimentation by allowing one to solve instances using certifiable numerical methods in lieu of more computationally complex symbolic methods. Additionally, the methods we use for proving the congruence modulo four and for producing a

Homotopy algorithms for solving structured determinantal systems

Multivariate polynomial systems arising in numerous applications have special structures. In particular, determinantal structures and invariant systems appear in a wide range of applications such as in polynomial optimization and related questions in real algebraic geometry. The goal of this thesis is to provide efficient algorithms to solve such structured systems. In order to solve the first kind of systems, we design efficient algorithms by using the symbolic homotopy continuation techniques. While the homotopy methods, in both numeric and symbolic, are well-understood and widely used in polynomial system solving for square systems, the use of these methods to solve over-detemined systems is not so clear. Meanwhile, determinantal systems are over-determined with more equations than unknowns. We provide probabilistic homotopy algorithms which take advantage of the determinantal structure to compute isolated points in the zero-sets of determinantal systems. The runtimes of our algorithms are polynomial in the sum of the multiplicities of isolated points and the degree of the homotopy curve. We also give the bounds on the number of isolated points that we have to compute in three contexts: all entries of the input are in classical polynomial rings, all these polynomials are sparse, and they are weighted polynomials. In the second half of the thesis, we deal with the problem of finding critical points of a symmetric polynomial map on an invariant algebraic set. We exploit the invariance properties of the input to split the solution space according to the orbits of the symmetric group. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in the number of points that we have to compute. Our results are illustrated by applications in studying real algebraic sets defined by invariant polynomial systems by the means of the critical point method