Uniform determinantal representations

Boralevi, Ada; van Doornmalen, Jasper; Draisma, Jan; Hochstenbach, Michiel E.; Plestenjak, Bor (2017). Uniform determinantal representations. SIAM journal on applied algebra and geometry, 1(1), pp. 415-441. SIAM 10.1137/16M1085656

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The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

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