Expansion of determinantal equations into polynomial form

Expansion methods for solving integral equations with multiple time lags using Bernstein polynomial of the second kind

In this paper, the Bernstein polynomials are used to approximate the solutions of linear integral equations with multiple time lags (IEMTL) through expansion methods (collocation method, partition method, Galerkin method). The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is car...

Bi-polynomial rank and determinantal complexity

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank ...

Determinantal Varieties over Truncated Polynomial Rings

We study higher order determinantal varieties obtained by considering generic m × n (m ≤ n) matrices over rings of the form F [t]/(t k), and for some fixed r, setting the coefficients of powers of t of all r × r minors to zero. These varieties can be interpreted as generalized tangent bundles over the classical deter-minantal varieties; a special case of these varieties first appeared in a prob...

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2×2 minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective space...

Roots of Bivariate Polynomial Systems via Determinantal Representations