A lifted square formulation for certifiable Schubert calculus

Abstract

Abstract. Formulating a Schubert problem as the solutions to a system of equations in either Plücker space or in the local coordinates of a Schubert cell usually involves more equations than variables. Using reduction to the diagonal, we previously gave a primal-dual formulation for Schubert problems that involved the same number of variables as equations (a square formulation). Here, we give a different square formulation by lifting incidence conditions which typically involves fewer equations and variables. Our motivation is certification of numerical computation using Smale's α-theory.A m × n matrix M with m ≥ n is rank-deficient if and only if all of its n × n minors vanish. This occurs if and only if there is a nonzero vector v ∈ C n with Mv = 0. There are m n minors and each is a polynomial of degree n in the mn entries of M. In local coordinates for v, the second formulation gives m bilinear equations in mn+n−1 variables, and the map (M, v) → M is a bijection over an open dense set of matrices of rank n−1. The set of rank-deficient matrices has dimension (m+1)(n−1), which shows that the second formulation is a complete intersection, while the first is not if m > n. The principle at work here is that adding extra information may simplify the description of a degeneracy locus.Schubert varieties in the flag manifold are universal degeneracy loci [6]. We explain how to add information to a Schubert variety to simplify its description in local coordinates. This formulates membership in a Schubert variety as a complete intersection of bilinear equations and formulates any Schubert problem as a square system of bilinear equations. This lifted formulation is both different from and typically significantly more efficient than the primal-dual square formulation of [8], as we demonstrate in Section 3.Our motivation comes from numerical algebraic geometry [15], which uses numerical analysis to represent and manipulate algebraic varieties on a computer. It does this by solving systems of polynomial equations and following solutions along curves. For numerical stability, low degree polynomials are preferable to high degree polynomials. More essential is that Smale's α-theory [14] enables the certification of computed solutions to square systems of polynomial equations [10], and therefore efficient square formulations of systems of polynomial equations are desirable. Furthermore, the estimates used in implementations of α-theory simplify for bilinear systems, as explained in [8, Rem. 2.11]. Interestingly, formulations as square systems of bilinear equations may also aid Gröbner basis 2010 Mathematics Subject Classification. 14N15, 14Q20.

What is numerical algebraic geometry?

“…The sixth article [45] considers problems in Schubert calculus, which are typically formulated as an overdetermined system. The authors show that these problems can be reformulated as a square system so that standard certification techniques based on Smale's α-theory can be employed.…”

Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm

“…To solve a Schubert problem on a computer requires that it is formulated as a system of polynomial equations in some coordinates. There are several formulations, including global Plücker coordinates, local Stiefel coordinates, and more exotic primal-dual [7] or lifted [10] coordinates. An advantage of local Stiefel coordinates is that they involve the fewest variables.…”

“…Figure 1. Stiefel coordinates corresponding to a checkerboard.n = 14 and k = 7, with permutation array π =(6,7,8,9,11,12,13,14,10,5,4,3,2,1). The entries 0 are forced by the requirement that the matrix be reduced.…”

Certification for Polynomial Systems via Square Subsystems

“…This requires computing an exact rational univariate representation [28] and using that to certify solutions. An alternate approach taken in [9,11] is to reformulate the system f , adding variables to obtain an equivalent square system, which is then used for certification.…”