A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

Hauenstein, Jonathan D.; Hein, Nickolas; Sottile, Frank

doi:10.1007/s10208-015-9270-z

Cited by 6 publications

(31 citation statements)

References 30 publications

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“…We first compare these formulations to the determinatal formulation of a particular Schubert variety. Next, we determine which of the two uses fewer added variables for each Schubert variety on a flag manifold in C 9 , and then compare their computational efficiency for solving three Schubert problems, including two from [8]. We almost always observe a gain in efficiency for the lifted formulation over the primal-dual formulation.…”

Section: Comparison With the Primal-dual Square Formulationmentioning

confidence: 96%

“…Even if redundant minors are eliminated, the number that remains will in general exceed |w|. This is discussed for Grassmannians in Section 1.3 of [8], where it is shown that after removing redundancy, |w| = 0, 1 or a 1 = 1, n−1 are the only cases for which this number of minors equals |w|. In Section 3.1 we present a typical example minimally requiring 17 minors, but where |w| = 4.…”

Section: Determinantal Formulation Of a Schubert Varietymentioning

confidence: 99%

“…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.…”

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confidence: 92%

“…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].…”

mentioning

confidence: 99%

“…In Section 2 we give our new lifted square formulation for Schubert varieties and Schubert problems, illustrating with some examples. In Section 3 we compare the efficiency of the lifted formulation with the primal-dual formulation of [8], demonstrating that the lifted formulation typically involves fewer equations and variables, and through three examples that computations using it consume fewer resouorces.…”

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A lifted square formulation for certifiable Schubert calculus

Hein

Sottile

2017

Journal of Symbolic Computation

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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.

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Section: Comparison With the Primal-dual Square Formulationmentioning

confidence: 96%

Section: Determinantal Formulation Of a Schubert Varietymentioning

confidence: 99%

mentioning

confidence: 92%

mentioning

confidence: 99%

mentioning

confidence: 99%

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